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\markboth{\hfil A singular ODE  \hfil EJDE--2000/12}
{EJDE--2000/12\hfil Luka Korkut, Mervan Pa\v si\'c, \& Darko \v Zubrini\'c \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~12, pp.~1--37. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  A singular ODE related to \\ quasilinear elliptic equations  
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35J60, 35B65, 34C10.
\hfil\break\indent
{\em Key words and phrases:} $p$-Laplacian, spherically symmetric, existence,
\hfil\break\indent
non-existence,  uniqueness, comparison principle, singular ODE, regularity.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted September 10, 1999. Published February 12, 2000.} }
\date{}
%
\author{Luka Korkut, Mervan Pa\v si\'c, \& Darko \v Zubrini\'c}
\maketitle

\begin{abstract} 
 We consider a quasilinear elliptic problem with the natural growth in the
 gradient.  Existence, non-existence, uniqueness, and qualitative properties of
 positive solutions are obtained.  We consider both weak and strong solutions.
 All results are based on the study of a suitable singular ODE of the first
 order.  We also introduce a comparison principle for a class of nonlinear
 integral operators of Volterra type that enables to obtain uniqueness of weak
 solutions of the quasilinear equation.  
 \end{abstract} 

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}

\def\wo#1#2#3{W^{#1,#2}_0(#3)}
\def\pd#1#2{\frac{\partial#1}{\partial#2}}
\def\od#1#2{\frac{d#1}{d#2}}
\def\qed{\hfill$\diamondsuit$\medskip}


\subsection{Introduction}

In this paper we consider a quasilinear elliptic problem and its
spherically symmetric, positive solutions in a ball, both in the
weak and strong sense. The main difficulty represents the presence of the
natural growth in the gradient on the right-hand side. We study
existence, non-existence, uniqueness and qualitative properties of
solutions. The quasilinear problem is studied by means of a
suitable singular ODE of the first order. To prove uniqueness of weak
solutions of quasilinear problem, we use a new type of comparison
principle for integral operators of Volterra type, recently
introduced by the second author. To our knowledge this seems to
be the first uniqueness result for quasilinear elliptic equations
with the natural growth in the gradient. Our existence proofs are
constructive in the sense that solutions possess explicit
integral representation, obtained by means of isoperimetric
equalities and monotone rearrangements.
Nonexistence results are obtained by
constructing an unbounded sequence of subsolutions. The results
seem to be new even in the case of $p=2$.

Relatively simple methods developed in this paper enable numerous
generalizations and variations. First, our existence results can
be used to study general quasilinear elliptic equations in
divergence form on arbitrary open and bounded set $\Omega$, see
\cite{paisop}. These methods can be
exploited in the study of quasilinear elliptic systems with
strong dependence in the gradient, in the study of biharmonic
equations, and even polyharmonic equations. It is also possible
to study the corresponding variational inequalities.

Our basic model is the following class of quasilinear elliptic
equations
with the natural growth in the gradient:
\begin{eqnarray}
&-\Delta_p v=\tilde g_0|x|^m+\tilde f_0|\nabla v|^p\quad\mbox{in
$B\setminus\{0\}$,}& \nonumber \\
&v=0 \quad\mbox{on $\partial B$,}& \label{pde}\\
&v(x) \mbox{ spherically symmetric and decreasing.}& \nonumber
\end{eqnarray}
Here $B=B_R(0)$ is the ball of radius $R$ in ${\mathbb R}^N$, $N\ge1$,
$1<p<\infty$, $\Delta_p v=\mathop{\rm div}(|\nabla v|^{p-2}\nabla v)$. 
We shall also need the conjugate exponent $p'=\frac p{p-1}$. Also, 
we denote ${\mathbb R}^+=[0,\infty)$.
We assume that the constants $\tilde f_0$ and $\tilde
g_0$ are positive real numbers, and $m\in {\mathbb R}$ can also be
negative, i.e.\ the right hand side of (\ref{pde}) may be
singular.

We can interpret solutions of equation (\ref{pde}) in three ways.
We say that $v(x)$ is

\begin{enumerate}
\item[(i)] a strong solution of (\ref{pde}) if $v\in
C^2(B\setminus\{0\})\cap C(\overline B)$;

\item[(ii)] a classical solution of (\ref{pde}) if $v\in C^2(\overline B)$;

\item[(iii)] a weak solution of (\ref{pde}) if $v\in \wo1p{B}\cap L^{\infty}(B)$, and
equation (\ref{pde}) is satisfied in the weak sense:
\begin{equation}\label{slabo}
\displaystyle\int_B|\nabla v|^{p-2}\nabla v\cdot\nabla\varphi\,dx=\tilde g_0\int_B|x|^m\varphi(x)\,dx+
\tilde f_0\int_B|\nabla v|^p\varphi(x)\,dx
\end{equation}
for all $\varphi\in\wo1pB\cap L^{\infty}(B)$.
\end{enumerate}

In Section~\ref{connS} we show that the study of (\ref{pde}) can be related
to the study of the following ordinary differential equation:
\begin{eqnarray}\label{ode}
&\od{\omega(t)}t=g_0\gamma
t^{\gamma-1}+f_0\frac{\omega(t)^\delta}{t^\varepsilon},\quad t\in(0,T),&\\
&\omega(0)=0,& \nonumber
\end{eqnarray}
where the constants $g_0$, $f_0$, $\delta$, $\gamma$, $T$ are assumed to
be positive, and $\varepsilon\in{\mathbb R}$. Note that for $\varepsilon>0$ problem
(\ref{ode}) is singular. In Section~\ref{weak} we study weak solutions of
(\ref{pde}), where we exploit techniques of isoperimetric equalities and
monotone rearrangements.


In Section~\ref{singularS} we are interested in finding sufficient conditions on pairs
$(f_0,g_0)$ of positive real numbers
that ensure existence of at least one solution $\omega$ of
(\ref{ode}),
i.e.\ of
(\ref{fixedpteq}).
While for $\delta\in(0,1)$ the problem is
solvable for all $(f_0,g_0)$, the case of $\delta>1$
is strikingly different. Namely, in the latter case we always
have an unbounded set of pairs $(f_0,g_0)$ for which the problem is
not solvable.
More precisely, we obtain two explicit positive constants
$C_1<C_2$ depending only on $\delta$, $\gamma$, $\varepsilon$, and $T$, such that if
$$
f_0\le \frac{C_1}{g_0^{\delta-1}}
$$
then ODE (\ref{ode}) is solvable, while for
$$
f_0\ge \frac{C_2}{g_0^{\delta-1}}
$$
problem (\ref{ode}), and even problem (\ref{odde}) below,
is not solvable. We know (almost) nothing
about
solvability of (\ref{ode}) in the case when
$$
\frac{C_1}{g_0^{\delta-1}}<f_0<\frac{C_2}{g_0^{\delta-1}}
$$
See Theorems~\ref{banach}, \ref{schauder},
\ref{monotone} for solvability results related to (\ref{ode}),
and Theorem~\ref{non} for
a nonsolvability result. In Section~\ref{qual1} we show a regularity
result for solutions of~(\ref{ode}) at singular point $t=0$.

It is interesting that in the case when $(f_0,g_0)$ belongs to
existence region described above, we can also
prove uniqueness of the solution, but only in the set of the form
(\ref{DKm}).
An important tool in obtaing uniqueness result
is played by a  pointwise
comparison principle for general operators of Volterra type, that
we introduce in Theorem \ref{compar}, see Section~\ref{comparS}.
Its first version has appeared in Pa\v si\'c \cite{pa}.

In Section \ref{quasi} we apply existence and non-existence
results from Section \ref{singularS} to study the problem of
existence, qualitative properties, and non-existence of solutions
of (\ref{pde}).
Using the above results it is  easy to obtain two positive
constants $\tilde C_1<\tilde C_2$ depending only on $m$, $p$,
$N$, and $R$,
such that if
$$
\tilde f_0\le \frac{\tilde C_1}{\tilde g_0^{p'-1}}
$$
then PDE (\ref{pde}) possesses a strong solution generated by a
solution of the corresponding singular ODE (\ref{ode}), that we
call $\omega$-solution. In fact, we
obtain that these strong solutions coincide with weak solutions,
and furthermore, (\ref{pde}) is uniquely solvable in the weak
sense, see Theorem~\ref{uniqqque}.
On the other hand, for
$$
\tilde f_0\ge \frac{\tilde C_2}{\tilde g_0^{p'-1}}
$$
problem (\ref{pde}) has neither weak nor strong solutions.
Again, we know (almost) nothing about solvability of (\ref{pde})
in the case when
$$
\frac{\tilde C_1}{\tilde g_0^{p'-1}}<\tilde f_0<\frac{\tilde
C_2}{\tilde g_0^{p'-1}}
$$
Existence results are supplied with constructive
proofs, and, as we have said, we have even uniqueness of weak
solutions.
See our main results
in Theorem~\ref{main} and Theorem~\ref{uniqqque} for precise
statements.
The non-existence result represents a
refinement of the corresponding result in  Pa\v si\'c~\cite{kan}.
Aplying regularity result from Section~\ref{qual1} we are able to
describe the behaviour of solutions of (\ref{pde}) at $x=0$ and on the
boundary of~$B$.

It is worth noting that  the phenomenon of having existence and
non-existence regions with respect to $(\tilde f_0,\tilde g_0)$
for (\ref{pde}) is due to the presence of
the term $\tilde f_0|\nabla v|^p$ with $\tilde f_0>0$, on the
right-hand side. Namely,
if we have $\tilde f_0=0$, we have no more this effect with
respect to parameter $\tilde g_0$.

%%%%%%%%%%%%%%%%%



\section{Connection between ODE and PDE}
\label{connS}

\subsection{Strong solutions}


Here we want to describe the connection between
strong solutions of quasilinear elliptic problem
(\ref{pde}) and solutions of the corresponding singular ODE
(\ref{ode}).
We obtain solutions of (\ref{ode}) as fixed points of the
following singular nonlinear
integral operator of Volterra type:
\begin{equation}\label{K}
K\,:\,D(K)\subset C([0,T])\to C([0,T]),\quad
K\varphi(t)=g_0t^{\gamma}+f_0\int_0^t\frac{\varphi(s)^\delta}{s^\varepsilon}\,ds.
\end{equation}
A domain $D(K)$ will be chosen so that the corresponding
fixed point equation
\begin{eqnarray}\label{fixedpteq}
&\omega\in D(K),\quad\omega=K\omega&
\end{eqnarray}
is solvable. We shall deal with two types of domains.
When we apply Banach's contraction method or
Schauder's fixed point theorem, then we shall use the domain
\begin{equation}\label{DK}
D(K)=\{\varphi\in C([0,T])\,:\,0\le\varphi(t)\le Mt^\gamma\},
\end{equation}
with a suitable constant $M>0$ independent of $\varphi$,
which ensures that $R(K)\subseteq
D(K)$, see Theorems \ref{banach} and \ref{schauder}
(by $R(K)$ we denote the range of $K$).
In the case of monotone iterations we take much larger domain,
see Theorem \ref{monotone}:
\begin{equation}\label{DKm}
D(K)=\{\varphi\in C([0,T])\,:\,\exists M_\varphi\ge0,\,\,0\le\varphi(t)\le M_\varphi
t^\gamma\}.
\end{equation}

It will be convenient to introduce an auxiliary
function $V\,:\,[0,|B|]\to{\mathbb R}$, where $|B|$ is the Lebesgue measure
of $B$, such that
\begin{equation}\label{vV}
v(x)=V(C_N|x|^N),
\end{equation}
where $C_N$ is the volume of the unit ball in ${\mathbb R}^N$.
In fact, (\ref{vV}) will have two r\^oles: if a function $v(x)$
 is given,
then it will serve to define $V(s)$, and if $V(s)$ is given, it
will define $v(x)$.
If $V(s)$ is decreasing, then (\ref{vV}) implies that
\begin{equation}\label{DpvV}
-\Delta_p v=C_N^{p/N}N^p\od{}s\left(s^{p(1-1/N)}\left|\od
Vs\right|^{p-1}\right)
\end{equation}


\paragraph{From solutions of ODE to strong solutions of PDE.}
Let $\omega\,:\,[0,T]\to{\mathbb R}$ be a solution of (\ref{ode}).
In order to obtain a strong solution of (\ref{pde}) via (\ref{vV}),
we define the function 
\begin{equation}\label{V}
V(s)=\int_s^T \frac{\omega(\sigma)^\beta}{\sigma^{\alpha}}\,d\sigma,\quad T=|B|,
\end{equation}
with $\alpha$ and $\beta$ specified below.
We shall always have that $0\le\omega(s)\le Ms^\gamma$ with some
$\gamma>0$ and $M>0$, so that  $V(0)<\infty$ provided
$\alpha<\beta\gamma+1$.
Using (\ref{V}) we obtain
\begin{equation}\label{Dpvo}
-\Delta_pv(x)=C_N^{p/N}N^p\od{}s\left(s^{p(1-\frac1N)-\alpha(p-1)}\omega(s)^{\beta(p-1)}
\right),\quad s=C_N|x|^N.
\end{equation}
An easy computation shows that we have the following relation
between $\omega(s)$ and $|\nabla v|$, see (\ref{vV}) and (\ref{V}):
\begin{equation}\label{ov}
\omega(s)=N^{-1/\beta}C_N^{\frac{\alpha-1}\beta}|x|^{\frac{N\alpha-N+1}\beta}|\nabla
v(x)|^{1/\beta},\quad s=C_N|x|^N.
\end{equation}

In the following lemma we generate strong solutions of (\ref{pde})
starting from solutions of the corresponding ODE (\ref{ode}),
so that the coefficients $\alpha$, $\beta$, $\gamma$, $\delta$, $\varepsilon$ are defined
by $\tilde f_0$, $\tilde g_0$, $m$, $p$, $N$ using (\ref{ab}),
(\ref{cde}) and (\ref{veza}).


 \begin{lemma}\label{conn}
 Let $\tilde f_0$ and $\tilde g_0$ be given positive real
 numbers. Assume that $1<p<\infty$,
\begin{equation}\label{mpN}
m>\max\{-p,-N\}.
\end{equation}
Let the constants $\alpha$, $\beta$, $\gamma$, $\delta$, and $\varepsilon$ be defined
by
\begin{equation}\label{ab}
\alpha=p'(1-\frac1N),\quad \beta=\frac{p'}p,
\end{equation}
and
\begin{equation}\label{cde}
\gamma=1+\frac mN,\quad\delta=p',\quad \varepsilon=p'(1-\frac1N),
\end{equation}
and let
\begin{equation}\label{veza}
g_0=\frac{\tilde g_0}{C_N^{\frac{m+p}N}N^{p-1}(m+N)},\quad
f_0=\tilde f_0.
\end{equation}
Then we have $\alpha<\beta\gamma+1$, $\delta>\frac{\varepsilon-1}\gamma+1$, $\gamma>0$, and
for any solution $\omega$ of (\ref{ode}) with $T=|B|$,
 such that $0\le \omega(t)\le
Mt^\gamma$ for some $M>0$, we have that the corresponding
 function $v(x)$ defined by (\ref{vV}) and (\ref{V}) is a
strong solution of quasilinear problem (\ref{pde}).
Furthermore, the following relation holds:
\begin{equation}\label{osn2}
u'(r)=-\frac N{C_N^{
\frac{p'}p(1-\frac pN)}}
r^{-\frac
{p'}p(N-1)}\,\omega(C_Nr^N)^{p'/p},
\end{equation}
where $u\,:\,[0,R]\to{\mathbb R}$ is defined by $u(r)=v(x)$, $r=|x|$.
\end{lemma}

\noindent{\sc Proof.}
Note that both the integrability condition $\alpha<\beta\gamma+1$ of
$\sigma^{-\alpha}\omega(\sigma)^\beta$ in
(\ref{V}) and the condition $\delta>\frac{\varepsilon-1}\gamma+1$
are equivalent to $m>-p$.
Also, since $m>-N$, then
 $\gamma=1+\frac mN>0$.

Using (\ref{ab}) we see that
 (\ref{Dpvo}) reduces to
\begin{equation}\label{odj}
-\Delta_p v=C_N^{p/N}N^p\,\od{\omega}s.
\end{equation}
Now let us take into account our singular ordinary differential
equation (\ref{ode})
with the values of $\gamma$, $\delta$, and $\varepsilon$ defined in
(\ref{cde}).
Substituting into (\ref{odj}) and using (\ref{ov})
we obtain that $v(x)$ is a strong solution
of (\ref{pde}):
\begin{eqnarray}
-\Delta_p v&=& C_N^{p/N}N^p[g_0\gamma s^{\gamma-1}+f_0s^{-\varepsilon}\omega(s)^\delta]\nonumber\\
&=&
C^{p/N}_NN^p[g_0\gamma(C_N|x|^N)^{\gamma-1}+f_0(C_N|x|^N)^{-\varepsilon}\omega(C_N|x|^N)^\delta]\\
&=&\tilde g_0|x|^m+\tilde f_0|\nabla v|^p.\nonumber
\end{eqnarray}
 Relation (\ref{osn2}) follows from (\ref{ov})
and the fact that $|\nabla v(x)|=-u'(r)$.
\qed


We say that $v(x)$ is $\omega$-solution of PDE  (\ref{pde}), if it is
a strong solution which can be obtained as
in Lemma \ref{conn}, using the solution $\omega(t)$ of ODE
(\ref{ode}) such that $0\le \omega(t)\le Mt^\gamma$ for some $M>0$.

Note that our ODE (\ref{ode}) is singular only for $\varepsilon>0$.
For $\varepsilon$ as in (\ref{cde}) this condition corresponds to the case when $N\ge2$
in (\ref{pde}); if $N=1$ in (\ref{pde}), then $\varepsilon=0$ in equation
(\ref{ode}).
 On the other hand, the right-hand side of our PDE (\ref{pde})
 has singularity at $x=0$
 provided $m<0$, which means that $\gamma<1$ in the corresponding ODE
 (\ref{ode}).



\paragraph{From strong solutions of PDE to solutions of ODE.}
It is not difficult to show that a strong solution $v$ of (\ref{pde}) generates a
solution of a suitable ODE defined by (\ref{odde}), without any initial
condition. We seek for solutions contained in the set
\begin{equation}\label{DN}
D^+=\{\varphi\in C([0,T])\,:\,\varphi(t)\ge0,\,\,\mbox{$\varphi$ nondecreasing}\}.
\end{equation}

\begin{lemma}\label{conno}
Let $v$ be a strong solution of (\ref{pde}), where $\tilde f_0>0$,
$\tilde g_0>0$, and $m$, $p$, $N$ satisfy (\ref{mpN}). Define $V(s)$ by
(\ref{vV}), and let
\begin{equation}\label{oV}
\displaystyle
\omega(s)=s^{p(1-\frac1N)}\left|\od Vs\right|^{p-1},\quad s\in(0,T),\quad
T=|B|.
\end{equation}
Then $\omega$ satisfies the following
ODE:
\begin{eqnarray}
&\od{\omega(t)}t=g_0\gamma
t^{\gamma-1}+f_0\frac{\omega(t)^\delta}{t^\varepsilon},\quad t\in(0,T),&
\label{odde}\\
&\omega\in D^+.& \nonumber
\end{eqnarray}
where $g_0$, $f_0$ are defined by (\ref{veza}), and constants
 $\gamma$, $\delta$, $\varepsilon$ are defined by
(\ref{cde}). Also, $V(s)$ can
be represented by (\ref{V}), where $\alpha$ and $\beta$ are defined by
(\ref{ab}).
\end{lemma}

\noindent{\sc Proof.} From $V(s)=v(x)$, $s=C_N|x|^N$ we obtain $|\nabla
v|=NC_N^{1/N}s^{1-\frac1N}\left|\od Vs\right|$. Using
(\ref{pde}),
 (\ref{odj}), and (\ref{veza}) it is easy to show that $\omega$
satisfies (\ref{ode}).
From (\ref{oV}),
and using the fact that $V(s)$ is decreasing,
 we see that $\od Vs=-s^{-\alpha}\omega(s)^\beta$. Integrating from $s$ to $T$
 we obtain (\ref{V}).
\qed

%%%%%%%%%%%%%%%%

\subsection{Weak solutions}

\label{weak}

Theorem~\ref{thw2} below shows that every weak solution of
(\ref{pde}) is $\omega$-solution of (\ref{pde}).
In fact, we consider a more general problem than (\ref{pde}):
\begin{eqnarray}
&-\Delta_p v=\tilde g(|x|,v)+\tilde f_0|\nabla v|^p\quad\mbox{in
$B\setminus\{0\}$,}&\nonumber\\
&v=0 \quad\mbox{on $\partial B$,}& \label{pdde}\\
&v(x) \mbox{ spherically symmetric and decreasing.}& \nonumber
\end{eqnarray}
We define the notion of  weak solution of this equation as a
function $v\in\wo1pB\cap
L^{\infty}(B)$ satisfying integral identity analogous to
(\ref{slabo}).
In what follows we  assume $\tilde g\,:\,{\mathbb R}^+\times{\mathbb R}^+\to{\mathbb R}^+$
is a Carath\'eodory function such that
\begin{equation}\label{tildeg}
\int_0^{|B|}\tilde g\left(\left(\frac
s{C_N}\right)^{1/N},\varphi(s)\right)\,ds<\infty,\quad\forall\varphi\in
C([0,|B|]),\,\,\varphi\ge0.
\end{equation}

\begin{theorem}\label{thw1} Assume that (\ref{tildeg}) holds,
and let $v$ be a weak solution of (\ref{pdde}).  Let us define
 $V(s)$,  $s\in[0,T]$, $T=|B|$ by (\ref{vV}) and the
function $w\,:\,[0,T]\to{\mathbb R}$ by
\begin{equation}\label{ww}
\omega(s)=\frac1{N^pC_N^{p/N}}e^{-\tilde f_0V(s)}\int_0^s\tilde
g\left(\left(\frac\sigma{C_N}\right)^{1/N},V(\sigma)\right)e^{\tilde f_0
V(\sigma)}d\sigma.
\end{equation}
Then the functions $\omega$ and $V$ satisfy the following system of ODE's:
\begin{eqnarray}
&\od\omega s=\frac1{N^pC_N^{p/N}}\tilde g\left(\left(\frac
s{C_N}\right)^{1/N},V(s)\right)+\tilde
f_0\frac{\omega(s)^{p'}}{s^{p'(1-\frac1N)}}\quad a.e.\,\,s\in(0,T),&\nonumber\\
&\od Vs = -s^{p'(-1+\frac1N)}\omega(s)^{p'/p}\quad a.e.\,\,s\in(0,T)&
\label{system-ode}\\
&\omega(0)=0,\quad\omega\in AC^+([0,T]),&\nonumber\\
&V(T)=0,\quad V\in AC^+([a,T]),\quad\forall a>0\,.&\nonumber
\end{eqnarray}
Furthermore, we have
\begin{equation}
\omega(s)\le\frac{e^{\tilde f_0V(0)}}{N^pC_N^{p/N}}\int_0^s\tilde
g\left(\left(\frac\sigma{C_N}\right)^{1/N},V(\sigma)\right)\,d\sigma,
\end{equation}
and
$v\in C^{\infty}(\overline B\setminus\{0\})$.
If there exists $M>0$ such that $\omega(s)\le Ms^{1+m/N}$,
$s\in[0,T]$, and $m>-p$, then also $v\in C(\overline B)$, and $v$ is the strong
solution of (\ref{pdde}).

We have a partial converse: if $\omega$ and $V$ are solutions of
(\ref{system-ode}) such that $0\le\omega(s)\le Ms^{1+m/N}$ for some
$M>0$, then the corresponding function $v(x)$ defined by
(\ref{vV}) is a strong solution of (\ref{pdde}).
\end{theorem}

In the special case of $\tilde g(r,\eta)=\tilde g_0 r^m$, for some fixed
$m\in{\mathbb R}$ and $\tilde g_0>0$, we obtain problem (\ref{pde}).

\begin{theorem}\label{thw2} Assume that $m>-N$, and let $v(x)$ be a weak
solution of (\ref{pde}). If we define $V(s)$, $s\in[0,T]$,
$T=|B|$, by (\ref{vV}), then the function $\omega\,:\,[0,T]\to{\mathbb R}^+$
defined by
\begin{equation}\label{o}
\omega(s)=\frac{\tilde g_0}{N^pC_N^{\frac{m+p}N}}e^{-\tilde f_0V(s)}\int_0^s
\sigma^{\frac mN}e^{\tilde f_0 V(\sigma)}d\sigma,
\end{equation}
satisfies equation (\ref{ode}), where the constants $\gamma$, $\delta$,
$\varepsilon$ are defined by (\ref{cde}), and $f_0$, $g_0$ by
(\ref{veza}). Also, we have that for all $s\in[0,T]$
\begin{equation}\label{oM}
\omega(s)\le Ms^\gamma,\quad M= g_0\cdot e^{\tilde f_0V(0)}.
\end{equation}
We have $v\in C^{\infty}(\overline B\setminus\{0\})\cap C(\overline B)$. If
in addition to the above hypotheses we assume $m>-p$, then $v\in
C(\overline B)$ and the
weak solution of (\ref{pde}) is also $\omega$-solution.
\end{theorem}

\noindent{\sc Proof of Theorem~\ref{thw2}.} This theorem follows easily
from Theorem~\ref{thw1}. Concerning the continuity of $v$ on $B$,
note that
(\ref{oM}) and (\ref{V}) imply that $V(s)$ is continuous at $s=0$, and therefore
$v(x)$ is continuous at $x=0$.
The partial converse can be proved similarly as in
Lemma~\ref{conn}.
\qed

Before proceeding to proof of Theorem~\ref{thw1} we compare
 Lemma~\ref{conno} with
the above theorem. On the one hand, it should be
noted that the function $\omega(s)$
defined by (\ref{o})
satisfies the same ODE (\ref{ode}) as the one defined by
(\ref{oV}) in Lemma~\ref{conno}.
On the other hand, in Theorem~\ref{thw1} we have that
weak solutions of (\ref{pde}) yield
$w\in AC^+([0,T])$ and the estimate (\ref{oM}), while
in Lemma~\ref{conno} strong solutions yield only $\omega\in D^+$.
As we see from Theorem~\ref{thw2},
any weak solution of (\ref{pde}) is $\omega$-solution, while for
strong solutions this is an open problem.

Before proving Theorem~\ref{thw1} we recall some very well known
results on Schwartz symmetrization.

\begin{lemma}\label{schwartz} Let $v\in\wo1pB\cap L^{\infty}(B)$ be a spherically
symmetric and decreasing function and let $V(s)$, $s\in[0,|B|]$,
 be defined by (\ref{vV}). Then we have:

(i) $V(s)=v^*(s)$,  where by definition
$v^*(s)=|\{t\ge0\,:\,\mu(t)>s\}|$, $\mu(t)=|\{x\in B\,:\,v(x)>t\}|$;
also $\mu(s)=V^{-1}(s)$, where $V^{-1}$ denotes the inverse
function of $V$;

(ii) $V\in W^{1,p}_{loc}(0,|B|)\cap C([0,|B|])$, and $V\in
AC([a,|B|])$ for all $a>0$.
\end{lemma}

\noindent{\sc Proof.} (i) follows easily from the fact that $v$ is
decreasing. For the proof of (ii) see for example Rakotoson, Temam \cite{rt}.
\qed

\noindent{\sc Proof of Theorem~\ref{thw1}.} We define the function
\begin{equation}\label{testf}
\varphi(x)=e^{\tilde f_0V(C_N|x|^N)}S_{t,h}(v(x))
\end{equation}
where $t\in(0,T)$, $h>0$, and
\begin{eqnarray}
S_{t,h}(\tau)=
\left\{
\begin{array}{ll}
0,& \mbox{for $\tau\le t$},\\
\frac1h(\tau-t),& \mbox{for $t<\tau\le t+h$},\\
1,& \mbox{for $\tau>t+h$}.
\end{array}
\right.
\end{eqnarray}
It is easy to see that $\varphi\in\wo1pB\cap L^{\infty}(B)$.
Now if we test (\ref{pdde}) with $\varphi$ and let $h\to0$
we obtain
\begin{equation}\label{-d/dt}
-\od{}t\int_{\{v>t\}}|\nabla v|^pdx=e^{-\tilde f_0t}
\int_{\{v>t\}}\tilde g(|x|,v)\,e^{\tilde f_0
v(x)}dx.
\end{equation}
Let us show that
\begin{eqnarray}
-\od{}t\int_{\{v>t\}}|\nabla
v|^pdx&=&N^pC_N^{p/N}\mu(t)^{p(1-\frac1N)}|\mu'(t)|^{1-p}\label{w1}\\
\int_{\{v>t\}}\tilde g(|x|,v)\,e^{\tilde f_0
v}dx&=&\int_0^{\mu(t)}\tilde g\left(\left(\frac
\sigma{C_N}\right)^{1/N},V(\sigma)\right)e^{\tilde f_0
V(\sigma)}d\sigma.\label{w2}
\end{eqnarray}

(a) To prove relation (\ref{w1}), we start with:
$$
-\od{}t\int_{\{v>t\}}|\nabla v|^pdx=\lim_{h\to0}\frac1h\int_{\{t<v\le
t+h\}}|\nabla v|^pdx.
$$
Now use (\ref{vV}) and pass to the generalized
spherical coordinates:
\begin{eqnarray}
\lefteqn{\int_{\{t<v\le t+h\}}|\nabla v|^pdx}\\
&=&C_N^pN^p\int_{\{t<v\le
t+h\}}|x|^{p(N-1)}\left|\od Vs(C_N|x|^N)\right|^pdx\nonumber\\
&=&C_N^{p+1}N^{p+1}\int_{\{t<V(C_Nr^N)\le
t+h\}}r^{p(N-1)}\left|\od Vs(C_Nr^N)\right|^pr^{N-1}dr\nonumber\\
&=&N^pC_N^p\int_{\mu(t+h)}^{\mu(t)}s^{p(1-\frac1N)}\left|\od
Vs\right|^pds\nonumber\\
&=&N^pC_N^{p/N}\int_t^{t+h}\mu(\sigma)^{p(1-\frac1N)}|\mu'(\sigma)|^{1-p}d\sigma,\nonumber
\end{eqnarray}
where we used the change of variables
$V(s)=\sigma$, $\od Vs=\frac1{\mu'(s)}$, and $\mu^{-1}=V$, see
Lemma~\ref{schwartz}(i).
This proves (\ref{w1}). Relation (\ref{w2}) is proved
in the same way.


(b) Using (\ref{-d/dt}), (\ref{w1}), (\ref{w2}), and (\ref{ww})
we obtain
$$
1=\mu(t)^{p(-1+\frac1N)}|\mu'(t)|^{p-1}\omega(\mu(t)).
$$
This implies
\begin{eqnarray}
\od Vs&=&\lim_{h\to0}\frac1h\int_{V(s)}^{V(s+h)}1^{p'/p}\,dt\nonumber\\
&=&\lim_{h\to0}\frac1h\int_{V(s)}^{V(s+h)}\mu(t)^{p'(-1+\frac1N)}|\mu'(t)|
\omega(\mu(t))^{p'/p}dt\nonumber\\
&=&-\lim_{h\to0}\frac1h\int_s^{s+h}\tau^{p'(-1+\frac1N)}\omega(\tau)^{p'/p}
d\tau\nonumber\\
&=&-s^{p'(-1+\frac1N)}\omega(s)^{p'/p}.\nonumber
\end{eqnarray}
Here we have used the change of variables $\mu(t)=\tau$, and the
fact that $\mu^{-1}(\tau)=V(\tau)$.

It is now easy to verify that the function $\omega(t)$ defined by
(\ref{ww}) satisfies the first ODE in (\ref{system-ode}).
Note that the second ODE in (\ref{system-ode}) implies that
$V(s)$ has the form (\ref{V}). From this it is easy to conclude
that
$\omega\in C^{\infty}((0,T])$, and then also $v\in C^\infty(\overline
B\setminus\{0\})$. If $\omega(s)\le Ms^\gamma$, then (\ref{V}),
(\ref{vV}) and $m>-p$ imply that $v\in C(\overline B)$.
\qed

%%%%%%%%%%%%%%%%%%%%


\section{Singular ODE}
\label{singularS}

\subsection{Comparison principle and uniqueness of
solutions\label{comparS}}

Let $(X,\le)$ be a partially ordered set, and assume that $K\,:\,D(K)\subset
X\to X$ is an
operator. We say that $u$  is a {\sl subsolution} of $K$ if
\begin{equation}\label{subsol}
u\in D(K)\quad \mbox{and}\quad u\le Ku.
\end{equation}
We say that $v$  is a {\sl supersolution} of $K$ if
\begin{equation}\label{supersol}
v\in D(K)\quad \mbox{and}\quad v\ge Kv.
\end{equation}
Also, the operator $K$ is said to be nondecreasing if for any
$\varphi,\psi\in D(K)$ the assumption $\varphi\le\psi$ implies that $K\varphi\le
K\psi$.

We say that an operator $K\,:\,D(X)\subset X\to X$ has {\sl (weak) comparison
property} if for any subsolution $u$ and any supersolution $v$ of $K$
we necessarily have $u\le v$ ($Ku\ge Kv$). Now we formulate the following
very simple uniqueness lemma.

\begin{lemma}\label{Unique} Let $K\,:\,D(K)\subset X\to X$ have (weak) comparison
property. Then the fixed point equation $\omega=K\omega$  possesses at
most one solution in $D(K)$.
\end{lemma}

\noindent{\sc Proof.}
Let $\omega_1,\omega_2\in D(K)$ be such that $\omega_1=K\omega_1$, $\omega_2=K\omega_2$.
Since $\omega_1$ is a subsolution, and $\omega_2$ is a supersolution of $K$,
then $\omega_1\le\omega_2$. Similarly $\omega_2\le\omega_1$, that is $\omega_1=\omega_2$.

If $K$ has
weak comparison property then we obtain
$K\omega_1\le K\omega_2$ and
$K\omega_2\le K\omega_1$, that is $\omega_1=K\omega_1=K\omega_2=\omega_2$.
\qed

Let us describe our basic example of operators having (weak)
comparison property.  Let $X=C([0,T])$ with  pointwise partial
ordering, and assume that
$k(s,\eta)\,:\,[0,T]\times{\mathbb R}\to{\mathbb R}$ is a Carath\'eodory function
(measurable with respect to $s$ for all $\eta$ and continuous
with respect to $\eta$ for a.e.\ $s$) such that the integral
operator of Volterra type
\begin{equation}\label{volterra}
K\,:\,D(K)\subset C([0,T])\to C([0,T]),\quad K\varphi(t)=\int_0^t k(s,\varphi(s))\,ds
\end{equation}
is well defined on a nonempty domain $D(K)$. We say that the
operator $K$ has  property $(m)$ if
for all $\varphi,\psi\in D(K)$, for all
$a\in[0,T)$ and for all $c\in(a,T)$, there exist $b\in(a,c)$ and 
$m(a,b,\varphi,\psi)\in[0,1)$ such that 
\begin{equation} \label{m}
\|k(\cdot,\varphi(\cdot))-k(\cdot,\psi(\cdot))\|_{L^1(a,b)}\le
m(a,b,\varphi,\psi)\cdot\|\varphi-\psi\|_{L^{\infty}(a,b)}.
\end{equation}
We shall also say that the function $k(s,\eta)$ has property
$(m)$.

It is easy to see that our $(m)$-condition for operator
$K$ will be satisfied if we can obtain the last inequality in (\ref{m})
so that the following property on the coefficient
$m(a,b,\varphi,\psi)$ is satisfied:
\begin{equation}\label{mm}
\forall \varphi,\psi\in D(K),\,\,\forall a\in[0,T),\,\,\lim_{b\to
a+}m(a,b,\varphi,\psi)=0.
\end{equation}
This will be the situation in all applications that follow, see
Lemma \ref{unique} below.
In fact, even more general condition suffices for $(m)$-condition
to be fulfilled:
\begin{equation}\label{m-liminf}
\forall \varphi,\psi\in D(K),\,\,\forall a\in[0,T),\,\,\liminf_{b\to
a+}m(a,b,\varphi,\psi)\in[0,1).
\end{equation}
However, we do not know any example of operator $K$ satisfying property
(\ref{m-liminf}) which does not satisfy (\ref{mm}).



The following comparison principle will play basic r\^ole in
obtaining uniqueness results for the fixed point problem $\omega=K\omega$
using monotone iterations method.

\begin{theorem}\label{compar} (comparison principle) Let $K\,:\,D(K)\subset C([0,T])\to C([0,T])$ be
an integral operator of Volterra type given by (\ref{volterra}),
satisfying property (m), and such that $R(K)\subseteq D(K)$.

\begin{description}
  \item{(a)} If $k(s,\cdot)$ is nondecreasing for a.e.\ $s$ then
  the operator $K$ has comparison property, that is, if $u,v\in D(K)$ are such
  that $u\le Ku$, $v\ge Kv$, then necessarily $u\le v$.

 \item{(b)} If $k(s,\cdot)$ is nonincreasing for a.e.\ $s$ then $K$
 has weak comparison property, that is, if $u,v\in D(K)$ are such
  that $u\le Ku$, $v\ge Kv$, then necessarily $K(u)\ge K(v)$.
\end{description}

In both cases the fixed point equation $\omega=K\omega$ possesses at most
one solution in $D(K)$.
\end{theorem}

\noindent{\sc Proof.}
(a) Let $u$ and $v$ be a subsolution and supersolution of $K$
respectively. Denote $\theta=Ku$ and $\omega=Kv$. Then we have
$\theta,\omega\in D(K)\cap AC([0,T])$, and
$u\le\theta$, $v\ge\omega$ on $[0,T]$. Since $k(s,\cdot)$ is
nondecreasing for a.e.\ $s$ we obtain the following two
differential inequalities
\begin{eqnarray}\label{diffin}
&\od{\theta}t\le k(t,\theta(t))\quad a.e.\ t\in[0,T],&\\
&\od{\omega}t\ge k(t,\omega(t))\quad a.e.\ t\in[0,T].&\nonumber
\end{eqnarray}
Let us first prove that
\begin{equation}\label{usp}
\theta(t)\le\omega(t)\quad\mbox{for all $t\in[0,T]$}.
\end{equation}
It suffices to show that
\begin{eqnarray}\label{fact}
\begin{array}{ll}
\mbox{$\forall a\in[0,T)$ such that $\theta(a)=\omega(a)$ there is no
$c\in(a,T]$}&\\
\mbox{such that $\forall t\in(a,c]\,\,\theta(t)>\omega(t)$}.
\end{array}
\end{eqnarray}
We argue by contradiction. Assume, contrary to (\ref{fact}),
that there exists $a\in[0,T)$ such that $\theta(a)=\omega(a)$ and let
there exist $c\in(a,T]$ satisfying $\theta(t)>\omega(t)$ for all
$t\in(a,c]$. Integrating (\ref{diffin}) over $[a,t]$ with
$t\in[a,c]$ and using property (m) we obtain
\begin{eqnarray*}
|\theta(t)-\omega(t)|&=&\theta(t)-\omega(t)\le\int_a^tk(s,\theta(s))\,ds-
\int_a^tk(s,\omega(s))\,ds \\
&\le&\int_a^t|k(s,\theta(s))-k(s,\omega(s))|\,ds\le
\|k(\cdot,\theta(\cdot))-k(\cdot,\omega(\cdot))\|_{L^1(a,c)}\\
&\le& m(a,c,\theta,\omega)\,\|\theta-\omega\|_{L^{\infty}(a,c)}.
\end{eqnarray*}
Taking the maximum over $t\in[a,c]$ we obtain $\|\theta-\omega\|_{L^{\infty}(a,c)}\le
 m(a,c,\theta,\omega)\,\|\theta-\omega\|_{L^{\infty}(a,c)}$, and from
this
$\theta\equiv\omega$ on $[a,c]$, which is a contradiction.
This proves (\ref{usp}).

Now using (\ref{usp}), (\ref{subsol}), and (\ref{supersol}) we
have that
\begin{equation}
u\le Ku=\theta\le\omega=Kv\le v
\end{equation}

The case (b) is treated in the same way as (a).
The uniqueness claim follows from Lemma~\ref{Unique}.
\qed

Note that in case (a) the operator $K$ is monotone, while in
case (b) the operator $-K$ is monotone.

It is clear that if a subsolution $u$ and a supersolution $v$ of
$K$ are
given in advance in the above theorem, then we can replace the
condition $R(K)\subseteq D(K)$ with $Ku,Kv\in D(K)$ only.

Now we would like to describe our basic example of operators
satisfying property $(m)$.
We shall need  the following elementary
inequality which will be useful in the sequel:
\begin{equation}\label{ineq}
|\varphi^\delta-\psi^\delta|\le\delta\max\{\varphi^{\delta-1},\psi^{\delta-1}\}
|\varphi-\psi|,
\end{equation}
where $\varphi,\psi\ge0$ and $\delta>0$.
This follows immediately from the mean value theorem applied to $F(t)=t^\delta$,
$t\ge0$.

\begin{lemma}\label{unique} Assume that $\delta>\frac{\varepsilon-1}\gamma+1$,
$\delta>0$,
$\gamma>0$, $\varepsilon\in{\mathbb R}$, $f_0\in{\mathbb R}$, and $g_0\in{\mathbb R}$. Let the
Volterra type operator $K$ be defined by (\ref{K}), with its domain
$D(K)$
contained in the set
$\{\varphi\in C([0,T])\,:\,\exists M_\varphi\ge0,\,\,0\le\varphi(t)\le M_\varphi
t^\gamma\}$.
Then the operator $K$ has property $(m)$. In particular, the
equation $\omega=K\omega$ possesses at most one solution in $D(K)$.
\end{lemma}

\noindent{\sc Proof.} Here we have
$$
k(s,\eta)=\gamma g_0 s^{\gamma-1}+f_0\frac{\eta^\delta}{s^\varepsilon}.
$$
For any $\varphi,\psi\in D(K)$, $0\le a<b\le T$, we have:
\begin{eqnarray}
\|k(\cdot,\varphi(\cdot))-k(\cdot,\psi(\cdot))\|_{L^1(a,b)}
&\le& |f_0|\int_a^b\frac{|\varphi(s)^\delta-\psi(s)^\delta|}{s^\varepsilon}\,ds\nonumber\\
&\le& |f_0|\delta
M^{\delta-1}\int_a^bs^{\gamma(\delta-1)-\varepsilon}|\varphi(s)-\psi(s)|\,ds
\nonumber\\
&\le& m(a,b,\varphi,\psi)\|\varphi-\psi\|_{L^{\infty}(a,b)}
\end{eqnarray}
where we have denoted
\begin{eqnarray}
&\displaystyle  M=\max\{M_\varphi,M_\psi\},&\nonumber\\
&\displaystyle m(a,b,\varphi,\psi)=\frac{|f_0|\delta M^{\delta-1}}{\gamma(\delta-1)-\varepsilon+1}
[b^{\gamma(\delta-1)-\varepsilon+1}-a^{\gamma(\delta-1)-\varepsilon+1}].&
\end{eqnarray}
The condition (\ref{mm}) is clearly satisfied. The uniqueness claim follows
from Theorem \ref{compar}.
\qed



The following lemma provides  some sufficient conditions for
$R(K)\subseteq D(K)$ to hold.

\begin{lemma}\label{RK} Let the operator  $K$ be defined by
(\ref{K}), where  $f_0$ and $g_0$ are given positive real
numbers, $\delta\ge\frac{\varepsilon-1}\gamma+1$,  $\gamma>0$, and $\varepsilon\in{\mathbb R}$.
\smallskip
(a) Assume that the domain $D(K)$ is defined by (\ref{DK}),
 and let $M>0$
be such that
\begin{equation}\label{source}
g_0\le M-f_0\frac{M^\delta T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}.
\end{equation}
Then $R(K)\subseteq D(K)$, and the function $v(t)=Mt^\gamma$ is a
supersolution of $K$.

(b) If $D(K)$ is defined by (\ref{DKm}), then
$R(K)\subseteq D(K)$.
\end{lemma}

\noindent{\sc Proof.} (a) Let us take any $\varphi\in D(K)$. Then
\begin{equation}\label{unif}
\displaystyle K\varphi(t)\le g_0t^\gamma+f_0M^\delta\int_0^ts^{\gamma\delta-\varepsilon}ds
\le g_0t^\gamma+\frac{f_0M^\delta}{\gamma\delta-\varepsilon+1}t^{\gamma\delta-\varepsilon+1}.
\end{equation}
Since $\gamma\delta-\varepsilon+1\ge\gamma$ and $t/T\le1$, we have $\left(\frac
tT\right)^{\gamma\delta-\varepsilon+1}\le\left(\frac tT\right)^\gamma$, and therefore
$$
0\le
K\varphi(t)\le\left(g_0+f_0\frac{M^\delta T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}\right)t^\gamma
\le Mt^\gamma,
$$
that is, $K\varphi\in D(K)$. The proof that $v(t)=Mt^\gamma$ is a
supersolution, that is, $Kv\le v$, can be obtained in the same
way.

(b) For any $\varphi\in D(K)$ we have in the same way as in~(a),
$$
0\le K\varphi(t)\le
\left(g_0+f_0M_\varphi^\delta\frac{T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}\right)t^\gamma.
$$
and therefore there exists $M>0$ such
that $K\varphi(t)\le Mt^\gamma$, that is, $K\varphi\in D(K)$.
\qed

%%%%%%%%%%%%%%




\subsection{Existence and uniqueness of solutions}

\subsubsection{Contraction method}
\label{contract}

In this section we present a constructive proof of
existence (and uniqueness) of solutions of (\ref{ode}) based on the
method of contraction.
It will be convenient to introduce the vector space $X_\gamma$
of all functions $\varphi\in C([0,T])$ such that
\begin{equation}
\|\varphi\|_\gamma=\sup_{t\in(0,T]}\frac{|\varphi(t)|}{t^\gamma}<\infty,
\end{equation}
where $\gamma>0$ is given. This norm is equal to
$\|\varphi\|_\gamma=\inf\{M>0\,:\,|\varphi(t)|\le Mt^\gamma\}$. It is not difficult to see that
$(X_\gamma,\|\cdot\|_\gamma)$ is a Banach space which is continuously
imbedded into $C([0,T])$.

\begin{theorem}\label{banach} Let $\delta\ge\frac{\varepsilon-1}\gamma+1$, $\delta>0$,
 $\gamma>0$,
$\varepsilon\in{\mathbb R}$.

(a) Let $M>0$ be given, and assume that $f_0$ and $g_0$ are
positive real numbers satisfying
conditions (\ref{source})  and
\begin{equation}\label{contraction}
f_0<\frac{\gamma\delta-\varepsilon+1}{\delta M^{\delta-1}T^{\gamma(\delta-1)-\varepsilon+1}}.
\end{equation}
Then the operator $K\,:\,D(K)\to D(K)$ given by (\ref{K}) and
(\ref{DK}) is  well
defined and $X_\gamma$-contractive. There
exists a unique $\omega\in D(K)$ such that $\omega=K\omega$. Furthermore, if
$\delta>\frac{\varepsilon-1}\gamma+1$, then the solution $\omega$ is unique in the set
defined by (\ref{DKm}).

(b) Let $\delta>1$. The set of all pairs $(f_0,g_0)$ of positive real
numbers for which there exists $M>0$ satisfying
conditions (\ref{source}) and (\ref{contraction}) is described by the following inequality:
\begin{eqnarray}\label{hypb}
f_0<\frac{\gamma\delta-\varepsilon+1}
{\delta T^{\gamma(\delta-1)-\varepsilon+1}(\delta'g_0)^{\delta-1}},
\end{eqnarray}
where $\delta'=\frac{\delta}{\delta-1}$.
Problem (\ref{K}) is solvable for all such $(f_0,g_0)$.
\end{theorem}

\noindent{\sc Proof.} (a) By Lemma \ref{RK} we have that $R(K)\subseteq D(K)$.
To show that $K$ is contraction, let $\varphi,\psi\in D(K)$. Then
using
inequality
(\ref{ineq}) we obtain
\begin{eqnarray}\label{Kf}
\frac1{t^\gamma}|K\varphi(t)-K\psi(t)|&=&\frac{f_0}{t^\gamma}\int_0^t
\frac{|\varphi(s)^\delta-\psi(s)^\delta|}{s^\varepsilon}\,ds\nonumber\\
&\le&\frac{f_0}{t^\gamma}\delta\int_0^t
\frac{(Ms^\gamma)^{\delta-1}}{s^{\varepsilon-\gamma}}\cdot
\frac{|\varphi(s)-\psi(s)|}{s^\gamma}\,ds\nonumber\\
&\le&\frac{f_0}{t^\gamma}\delta M^{\delta-1}\|\varphi-\psi\|_\gamma\int_0^t
s^{\gamma\delta-\varepsilon}ds\\
&\le& c(f_0)\|\varphi-\psi\|_\gamma,\nonumber
\end{eqnarray}
where
\begin{equation}\label{f00}
\textstyle c(f_0)=
f_0\delta M^{\delta-1}\frac{T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}<1
\end{equation}
because of (\ref{contraction}). Taking the supremum in (\ref{Kf}) over
$t\in(0,T]$ we obtain that $\|K\varphi-K\psi\|_\gamma\le
c(f_0)\|\varphi-\psi\|_\gamma$, which proves that $K$ is contraction.
The claim follows from Banach's fixed point theorem.
The uniqueness claim in the set defined by (\ref{DKm}) follows
from Lemma~\ref{unique}.

(b)
Let us fix any $M>0$, and consider the set $Q_M$
of all pairs $(f_0,g_0)$ of positive real numbers satisfying  conditions
(\ref{source}) and (\ref{contraction}). Owing to claim (a) that
we have just proved,
it suffices to show that the
set
\begin{equation}\label{Mb}
{\cal M}_b=\bigcup_{M>0}Q_M
\end{equation}
is described by (\ref{hypb}).  It is not difficult to see that for
any $(f_0,g_0)\in{\cal M}_b$ there exists $M>0$ such that
\begin{equation}\label{g0}
g_0= M-f_0\frac{M^\delta T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}
\end{equation}
and
\begin{equation}\label{f0}
f_0=\frac{\gamma\delta-\varepsilon+1}{\delta M^{\delta-1}T^{\gamma(\delta-1)-\varepsilon+1}}.
\end{equation}
Namely, the envelope of the family of lines (\ref{g0}) in
$(f_0,g_0)$-plain is the convex curve
defined by
(\ref{hypss}) below, as we shall see in the proof of Theorem
\ref{schauder}. The union of its tangents obviously contains
${\cal M}_b$.
Substituting (\ref{f0}) into (\ref{g0}) we get
$M=\frac{\delta}{\delta-1} g_0$. Inserting
this into (\ref{f0}) we obtain
\begin{equation}\label{hypbb}
f_0=\frac{\gamma\delta-\varepsilon+1}{\delta T^{\gamma(\delta-1)-\varepsilon+1}(\delta'g_0)^{\delta-1}},
\end{equation}
which proves the claim. Note that we have strict inequality in
(\ref{hypb}) because of the strict inequality in
(\ref{contraction}). Although the sets $Q_M$ are not open, their
union ${\cal M}_b$ is open.
\qed

This result extends the corresponding result in Pa\v
si\'c~\cite{pa}.
Note that we could also have used contraction method in the
larger space
$C([0,T])$ instead of $X_\gamma$, but in this case we obtain a weaker
result. Namely in this case we need
$\delta>\frac{\varepsilon-1}\gamma+1$, and the numerator of the right-hand side of
(\ref{contraction}) should be
$\gamma\delta-\varepsilon+1-\gamma$ instead of $\gamma\delta-\varepsilon+1$.

%%%%%%%%%%%%%%%%


\subsubsection{Schauder's method}

In this section we apply Schauder's fixed point theorem to obtain
existence result for solutions of our singular ODE (\ref{ode}).

\begin{lemma}\label{compact}  Let $f_0$ and $g_0$ be positive
real numbers.
Assume that $\delta>\frac{\varepsilon-1}\gamma$,
$\gamma>0$, $\varepsilon\in{\mathbb R}$,
and let the operator $K$ be defined by (\ref{K}),
with domain  (\ref{DK}) for some fixed
$M>0$. Then then the operator $K$ is compact with respect to
uniform topology.
\end{lemma}

\noindent{\sc Proof.}
It suffices to show that $R(K)$ is relatively compact
in $C([0,T])$.
We use Ascoli's theorem. To show equicontinuity of the family
$R(K)$ take any $K\varphi\in R(K)$. Then for any $a$, $b$ such that $0\le a<b\le
T$ we have
\begin{eqnarray}
|K\varphi(b)-K\varphi(a)|&\le&g_0(b^\gamma-a^\gamma)+f_0\int_a^b M^\delta s^{\gamma\delta-\varepsilon}ds\nonumber\\
&\le&g_0(b^\gamma-a^\gamma)+\frac{f_0 M^\delta}{\gamma\delta-\varepsilon+1}[b^{\gamma\delta-\varepsilon+1}-a^{\gamma\delta-\varepsilon+1}].
\end{eqnarray}
The last expression is equal to $h(b)-h(a)$, where
$h(t)=g_0t^\gamma+\frac{f_0M^\delta}{\gamma\delta-\varepsilon+1}t^{\gamma\delta-\varepsilon+1}$ is
uniformly continuous on $[0,T]$. Therefore
$|K\varphi(b)-K\varphi(a)|$ tends
 to zero uniformly
as $b-a\to0$.
Also, the the family of functions from $R(K)$ is uniformly
bounded, see (\ref{unif}):
$$
0\le K\varphi(t)\le
g_0T^\gamma+\frac{f_0M^\delta}{\gamma\delta-\varepsilon+1}T^{\gamma\delta-\varepsilon+1}.
$$
This proves that the operator $K$ is compact.
\qed

The following theorem shows that (\ref{ode}) is solvable also in the
case when we have equality in (\ref{hypb}).

\begin{theorem}\label{schauder} Assume that
$\delta\ge\frac{\varepsilon-1}\gamma+1$, $\delta>0$, $\gamma>0$, and $\varepsilon\in{\mathbb R}$.

(a) Let $f_0$ and $g_0$ be
positive real numbers
and let
$M>0$ satisfy (\ref{source}). Then there exists at least one
$\omega\in D(K)$ such
that $\omega=K\omega$, where $D(K)$ is defined by~(\ref{DK}).

(b) Assume that also $\delta>1$.
The set of all pairs $(f_0,g_0)$ of positive real
numbers for which there exists $M>0$ satisfying (\ref{source})
is described by the following inequality:
\begin{equation}\label{hyps}
f_0\le\frac{\gamma\delta-\varepsilon+1}{\delta T^{\gamma(\delta-1)-\varepsilon+1}(\delta'g_0)^{\delta-1}}.
\end{equation}
In particular, equation (\ref{ode}) is solvable for
all such $(f_0,g_0)$.

(c) If we assume that $\delta<1$, then for
each pair $(f_0,g_0)$ of positive real numbers there exists a
solution $\omega\in D(K)$ of the fixed point equation
$\omega=K\omega$, where $D(K)$ is defined by (\ref{DK}) with $M$ large
enough.

(d) If $\delta>\frac{\varepsilon-1}\gamma+1$, then the
solution $\omega$ in (a), (b), and (c) is unique in the domain $D(K)$ defined by
(\ref{DKm}).
\end{theorem}

\noindent{\sc Proof.} (a) Note that the domain $D(K)$ is
bounded, closed, and convex. By Lemma
\ref{compact} the operator $K$ is compact, and by Lemma \ref{RK}
we have $R(K)\subseteq D(K)$.
Existence of at least one solution $\omega\in D(K)$ of equation $\omega=K\omega$
follows from Schauder's fixed point theorem. For
$\delta>\frac{\varepsilon-1}\gamma+1$ we can give a constructive
proof of this result based on the method of monotone iterations,
see the remark after the proof.

(b) Let us denote the set of all  $(f_0,g_0)$ satisfying
(\ref{source}) by $T_M$. We have to show that the union of
these sets
\begin{equation}\label{Ms}
{\cal M}_s=\bigcup_{M>0}T_M
\end{equation}
is described by (\ref{hyps}). Consider the family of lines
\begin{equation}\label{hypsM}
g_0= M-f_0\frac{M^\delta T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1},\quad M>0.
\end{equation}
in the $(f_0,g_0)$--plane.  To find the envelope of this family,
we differentiate with respect to $M$:
$$
0=1-f_0\frac {\delta M^{\delta-1}T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}.
$$
Eliminating $M$ from the last two equations we obtain after
some elementary computation that
\begin{equation}\label{hypss}
f_0=\frac{\gamma\delta-\varepsilon+1}{\delta T^{\gamma(\delta-1)-\varepsilon+1}(\delta'g_0)^{\delta-1}}.
\end{equation}
Since the corresponding set of $(f_0,g_0)$ satisfying
(\ref{hyps}) is convex, we
obviously have that it is equal to ${\cal M}_s$.

(c) First, let us define $D(K)$ by (\ref{DK}), with $M$ to be chosen
later. The operator $K\,:\,D(K)\to C([0,T])$ is compact, see
Lemma~\ref{compact}. For any $\varphi\in D(K)$ we have
\begin{eqnarray}
\frac{K\varphi(t)}{t^\gamma}&=&g_0+\frac{f_0}{t^\gamma}\int_0^t\frac{\varphi(s)^\delta}{s^\varepsilon}\,ds\le
g_0+\frac{f_0M^\delta}{t^\gamma}\int_0^ts^{\gamma\delta-\varepsilon}ds\nonumber\\
&\le&g_0+f_0\frac{T^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta-\varepsilon+1}M^\delta.\nonumber
\end{eqnarray}
If $\delta<1$, we conclude that the last expression is $\le M$ for $M$ large
enough, that is, $K\varphi\in D(K)$. This shows that $R(K)\subset D(K)$.
The set $D(K)$ is bounded, closed, and convex in $C([0,T])$, and
the existence of a solution follows from Schauder's fixed point theorem.

(d) The uniqueness claim follows from Lemma~\ref{unique}.
\qed


Note that the existence (and uniqueness!) result that we obtained
via Schauder's
theorem is not constructive. However, if $\delta>\frac{\varepsilon-1}\gamma+1$
it is possible to
give a constructive proof. Namely, if we take $v(t)=Mt^\gamma$, with
$M$ as in (\ref{source}), then $v$ is a supersolution of $K$, see
Lemma \ref{RK}, and the corresponding operator $K$ has
(m)-property, see Lemma~\ref{unique}.
This implies the existence of the solution of
$\omega=K\omega$ using the method of monotone iterations in the same way
as in the proof of Theorem \ref{monotone} below.

Note that if we have strict inequality in (\ref{hyps}), then
we are in the situation of Theorem~\ref{banach}.
It is clear that
$\overline {\cal M}_b\subset{\cal M}_s$, the closure being taken in $(0,\infty)^2$. Here ${\cal M}_b$ and ${\cal M}_s$ are defined by
(\ref{hypb}) and (\ref{hyps}).

%%%%%%%%%%%%%%%%%%


\subsubsection{Monotone iterations}

In this section we apply the method of monotone iterations to
obtain existence of solutions of (\ref{ode}). First of all, note
that since $f_0$ and $g_0$ are positive, then the zero function $u=0$ is a
subsolution of integral operator $K$ defined by (\ref{K}).

\begin{theorem}\label{monotone}  Assume that
$\delta>\frac{\varepsilon-1}\gamma+1$, $\delta\ge\max\{1,\varepsilon/\gamma\}$, $\delta>\varepsilon$, $\gamma>0$,
$\varepsilon\in{\mathbb R}$,
and let us define
\begin{equation}\label{v}
z(t)=\gamma g_0e^{h_0
t^{1-\varepsilon/\delta}}\int_0^ts^{\gamma-1}e^{-h_0s^{1-\varepsilon/\delta}}ds,\quad
h_0=\frac {f_0}{1-\varepsilon/\delta}.
\end{equation}
If  $f_0$ and $g_0$ are positive real numbers such that
\begin{equation}\label{sourcem}
g_0\le T^{\frac\varepsilon\delta-\gamma}e^{-h_0T^{1-\varepsilon/\delta}},
\end{equation}
then $z\in D(K)$, where $D(K)$ is defined by
(\ref{DKm}), and $z$ is a supersolution of $K$: $z\ge Kz$.

There exists the unique solution $\omega\in D(K)$ of $\omega=K\omega$.
It can be obtained
constructively using the method of monotone iterations, and
the following estimate holds:
\begin{equation}\label{o<v}
g_0t^\gamma+f_0g_0^\delta\frac{t^{\gamma\delta-\varepsilon+1}}{\gamma\delta-\varepsilon+1}\le\omega(t)\le
z(t),\quad \forall t\in[0,T].
\end{equation}
\end{theorem}

\noindent{\sc Proof.} 
(a) We have that $v(t)\le C_1\int_0^ts^{\gamma-1}\,ds$, where
$C_1$ is a positive constant, which implies that
$z\in D(K)$. To prove that $z$ is a supersolution of $K$, note that
it satisfies the following linear ODE:
$z'(t)=g_0\gamma t^{\gamma-1}+f_0\frac{z(t)}{t^{\varepsilon/\delta}}$, $z(0)=0$, that is,
$$
z(t)=g_0t^\gamma+f_0\int_0^t\frac{z(s)}{s^{\varepsilon/\delta}}\,ds.
$$
Therefore, to achieve $z\ge Kz$ it suffices to have
$$
\frac{z(s)}{s^{\varepsilon/\delta}}\ge\frac{z(s)^\delta}{s^\varepsilon}=
\left(\frac{z(s)}{s^{\varepsilon/\delta}}\right)^\delta,\quad\forall
s\in[0,T].
$$
Since $\delta\ge1$, this is equivalent to
$$
\frac{z(s)}{s^{\varepsilon/\delta}}\le1.
$$
Now this inequality can be proved using the following easy consequence
of (\ref{v})
$$
z(t)\le\gamma g_0e^{h_0
t^{1-\varepsilon/\delta}}\int_0^ts^{\gamma-1}ds,
$$
and (\ref{sourcem}):
$$
\frac{z(t)}{t^{\varepsilon/\delta}}\le
 g_0t^{\gamma-\varepsilon/\delta}e^{h_0 t^{1-\varepsilon/\delta}}
\le g_0T^{\gamma-\varepsilon/\delta}e^{h_0 T^{1-\varepsilon/\delta}}\le1.
$$

(b) To prove existence, we proceed with the usual monotone
iterations scheme. Let the sequence $(u_k)$ in $D(K)$ be defined
by $u_k=Ku_{k-1}$, $u_0=0$, where we note that $R(K)\subseteq D(K)$ by
Lemma
\ref{RK}. Since $0\le z$,  $K$ is nondecreasing, and
$z$ is a supersolution,
it is easy to conclude that
 the sequence $u_k$ is monotone and
bounded in $C([0,T])$:
$$
u\le u_k\le z,\quad u_{k-1}\le u_k.
$$
Therefore, applying Dominated Convergence Theorem we conclude that
there exists $\omega\in L^{\infty}(0,T)$ such that $u_k\to \omega$ in
$L^1(0,T)$ as $k\to\infty$. Now using the continuity property
 of the
integral operator (that is, the
Lebesgue Dominated Convergence Theorem again) we easily see that
$$
K\omega=K(\lim_{k\to\infty} u_k)=\lim_{k\to\infty}Ku_k
=\lim_{k\to\infty}u_{k+1}=\omega,
$$
with the limits being taken in $L^1(0,T)$. The integral equality
$\omega=K\omega$ implies that $\omega\in C([0,T])$, and the claim is proved.
It is clear that $u_k\le\omega\le z$ for all $k$.
The estimate (\ref{o<v}) is obtained for $k=2$.

(c) The uniqueness result follows immediately from Lemma \ref{unique}.
\qed

As we have seen in Lemma \ref{RK},  the operator $K$ possesses
another supersolution $z(t)=Mt^\gamma$ for $(f_0,g_0)\in{\cal M}_s$.
If $\delta>\frac{\varepsilon-1}\gamma+1$, then as in the proof of the preceding theorem,
see step (b),
we can show that the solution in Theorem~\ref{schauder}
can be obtained constructively using monotone iterations.
Note that we obtained in fact two solutions of $\omega=K\omega$, using
Schauder's method and the method of monotone iterations. They
coincide due to Lemma~\ref{unique}, and therefore we have
that the a~priori estimate (\ref{o<v}) holds also for the solution
that we have obtained in Theorem~\ref{schauder}.

It is clear that the set ${\cal M}_s\setminus{\cal M}_m$ is always nonempty,
where ${\cal M}_s$ is defined as the set of all pairs $(f_0,g_0)$ of
positive numbers satisfying (\ref{hyps}), and  ${\cal M}_m$ is defined by
(\ref{sourcem}).
It is interesting that in some cases
 the set ${\cal M}_m\setminus {\cal M}_s$ can also be nonempty.
Indeed, let us fix $f_0>0$. Denote
by $g_s$ the maximum of all $g_0$ satisfying
(\ref{hyps}), and denote the right-hand side of (\ref{sourcem}) by $g_m$. Then
 we obtain that
$$
\left(\frac{g_s}{g_m}\right)^{\delta-1}=
\frac{(\delta-1)^{\delta-1}}{\delta^\delta}\,(\gamma\delta-\varepsilon+1)\,
\frac{\exp\left(\frac{\delta-1}{1-\varepsilon/\delta}f_0T^{1-\varepsilon/\delta}\right)}
{f_0T^{1-\varepsilon/\delta}}.
$$
Using the following elementary inequality $e^x\ge ex$ with
$x=\frac{\delta-1}{1-\varepsilon/\delta}f_0T^{1-\varepsilon/\delta}$
(the equality is
achieved only for $x=1$), we get
$$
\left(\frac{g_s}{g_m}\right)^{\delta-1}\ge
e\left(1-\frac1\delta\right)^\delta\frac{\gamma\delta-\varepsilon+1}{1-\varepsilon/\delta}
$$
Note that $e(1-1/\delta)^\delta<1$ for all $\delta>1$.
The remaining fraction on the right hand side is $>1$, but it can
be made as close to $1$ as we wish by fixing $\delta>1$,
taking $\gamma>0$ small enough, and then by taking $\varepsilon\ge0$ small enough,
so that all conditions of Theorem \ref{monotone} still
hold. The above equality condition $x=1$ is equivalent to
$T=\left(\frac{\delta-\varepsilon}{\delta(\delta-1)}\right)^{\delta/(\delta-\varepsilon)}$. Using this $T$
 we can therefore achieve that
$$
\left(\frac{g_s}{g_m}\right)^{\delta-1}=
e\left(1-\frac1\delta\right)^\delta\frac{\gamma\delta-\varepsilon+1}{1-\varepsilon/\delta}<1,
$$
that is, $g_m>g_s$ in this case.

%%%%%%%%%%%%%%%%%%%%%%


\subsection{Nonexistence of solutions}

The aim of this section is to show that problem (\ref{ode}), and
even (\ref{odde}),
is not solvable provided $f_0$ and $g_0$ are sufficiently large.


\begin{theorem}\label{non} Assume that $\delta>\frac{\varepsilon-1}\gamma+1$, $\delta>1$,
$\gamma>0$, and $\varepsilon\in{\mathbb R}$. Let $f_0$ and $g_0$ be positive real
numbers.
Assume that
\begin{eqnarray}\label{hypn}
f_0\ge\left\{
\begin{array}{ll}
\displaystyle\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^{\delta'}}
{(\delta-1)T^{\gamma(\delta-1)-\varepsilon+1}g_0^{\delta-1}}& \mbox{for $\varepsilon<1$,}\\
\displaystyle\frac{\gamma\,\delta^{\delta'}}
{T^{\gamma(\delta-1)-\varepsilon+1}g_0^{\delta-1}}& \mbox{for $\varepsilon\ge1$.}
\end{array}
\right.
\end{eqnarray}
Then problem  (\ref{ode}), and even (\ref{odde}), has no solutions in $D^+$, see
(\ref{DN}).
Furthermore, there exists a sequence of subsolutions
of $K$ which is unbounded in $C([0,T])$.
\end{theorem}

To prove this non-existence result, we state two auxiliary
propositions.

\begin{prop}\label{Ozm} Let $\delta>\frac{\varepsilon-1}\gamma+1$, $\delta>1$, $\gamma>0$,
and $\varepsilon\in{\mathbb R}$, and let us define a sequence of functions
$z_m(t)$ inductively by
\begin{equation}\label{zmv}
\displaystyle z_{m+1}(t)=f_0\int_0^t
\frac{z_m(s)^\delta}{s^\varepsilon}\,ds,\quad z_0(t)=g_0t^\gamma.
\end{equation}
Then for each solution $\omega\in D^+$  of (\ref{odde})  we have
\begin{equation}\label{ozm}
\omega(t)\geq \sum_{m=0}^{n}z_m (t),\ \ t\in [0,T],\ \
\forall n\in{\bf N}.
\end{equation}
Furthermore,
\begin{equation}\label{zm}
z_m (t)=\frac{g_0^{\delta^{m}}f_0^{\sum^{m-1}_{k=0}\delta^{k}}
t^{(1-\varepsilon)\sum^{m-1}_{k=0}\delta^{k}+\gamma\delta^m}}
{\prod^{m}_{k=1}[(1-\varepsilon)\sum^{k-1}_{j=0}\delta^{j}+\gamma\delta^k]^{\delta^{m-k}}}\geq
0,
\end{equation}
and $z_m\in D(K)$, with $D(K)$ defined by (\ref{DKm}).
The  function $\omega_n$ defined as the right-hand side of (\ref{ozm})
is a subsolution of $K$ for each $n=0,1,2\dots$
\end{prop}


 \begin{prop}\label{div}
 Let  $\delta>\frac{\varepsilon-1}\gamma+1$, $\delta>1$, $\gamma>0$, $\varepsilon\in{\mathbb R}$, and let
 $z_m$ be defined by (\ref{zm}).
Assume that condition (\ref{hypn}) is fulfilled, and let us define
\begin{eqnarray}\label{t*}
t^*:=\left\{
\begin{array}{ll}
\displaystyle\left(\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^{\delta'}}
{(\delta-1)f_0g_0^{\delta-1}}\right)^{1/[\gamma(\delta-1)-\varepsilon+1]}& \mbox{for
$\varepsilon<1$,}\\
\displaystyle\left(\frac{\gamma\,\delta^{\delta'}}
{f_0g_0^{\delta-1}}\right)^{1/[\gamma(\delta-1)-\varepsilon+1]}& \mbox{for
$\varepsilon\ge1$.}
\end{array}
\right.
\end{eqnarray}
Then $t^*\le T$ and $\sum_{m=0}^{\infty}z_m(t)=\infty$ for all $t\in[t^*,T]$.
\end{prop}

\noindent{\sc Proof of Theorem \ref{non}.}
We argue by contradiction. Assume that there exists a solution
$\omega\in D^+$ of (\ref{ode}).
Using Proposition \ref{Ozm} and inequality (\ref{ozm})
we obtain that $\sum_{m=0}^{\infty}z_m(t)\le \omega(t)<\infty$ for all $t\in[0,T]$
and all $m$.
But this contradicts Proposition \ref{div}.
Propositions \ref{Ozm} and \ref{div} imply that
$\omega_n$ is an unbounded sequence of subsolutions of $K$.
\qed

\noindent{\sc Proof of Proposition \ref{Ozm}.} We proceed by induction with
respect to $n$. Since $\omega(t)=K\omega(t)\ge g_0t^\gamma$, we
can use (\ref{ode}) to obtain
$$
\od\omega t\ge g_0\gamma t^{\gamma-1}+f_0g_0^\delta t^{\gamma\delta-\varepsilon}.
$$
Integrating over $(0,t)$, $t\in(0,T]$, we get
(recall a well known fact
that for $\omega$ nondecreasing we have $\omega(t)-\omega(0)\ge\int_0^t\od
\omega t\,dt$):
$$
\omega(t)\ge z_0(t)+ z_1(t),
$$
which proves the claim for $n=1$.

Assume the claim holds for some $n\in{\bf N}$ and let us prove that
it holds also for $n+1$. From (\ref{ode}), $\omega\in D^+$
and using $(a+b)^{\delta}\geq a^{\delta}+b^{\delta}$, $a\geq 0$,
$b\geq 0$, $\delta>1$, we obtain:
        \begin{displaymath}
        \frac{d\omega}{dt}=g_0\gamma t^{\gamma-1}+f_0 t^{-\varepsilon}\omega^{\delta}(t)\geq
        g_0\gamma t^{\gamma-1}+f_0 t^{-\varepsilon}\left(\sum_{m=0}^{n}z_m
        (t)\right)^{\delta}\ge
        \end{displaymath}
        \begin{displaymath}
        \geq g_0\gamma t^{\gamma-1}+f_0t^{-\varepsilon}\sum_{m=0}^{n}z_m^{\delta}(t).
	\end{displaymath}
        Integrating the preceding inequality over $(0,t)$, we obtain
        \begin{displaymath}
        \omega(t)\geq z_0(t)+\sum_{m=0}^nz_{m+1}(t)
        =\sum_{m=0}^{n+1}z_m (t),\ \ t\in [0,T].
	\end{displaymath}
This proves (\ref{ozm}).
Formula (\ref{zm}) can be checked easily by induction using
(\ref{zmv}).
To show that $z_m\in D(K)$, note that
$\delta>1$ and $\sum_{k=0}^{m}\delta^{k}=\frac{\delta^{m+1}-1}{\delta-1}$ imply:
        \begin{equation}\label{zmm}
        \displaystyle
         z_m (t)=\frac{g_0^{\delta^{m}}f_0^{\frac{\delta^{m}-1}{\delta -1}}
         t^{\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^m+\varepsilon-1}{\delta-1}}(\delta-1)^{\sum_{k=1}^m\delta^{m-k}}}
         {\prod^{m}_{k=1}[(\gamma(\delta-1)-\varepsilon+1)\delta^k+\varepsilon-1]^{\delta^{m-k}}}
        \end{equation}
Since $\gamma(\delta-1)-\varepsilon+1>0$ and $\delta^m>1$, we see that
the exponent at $t$ is $\ge\gamma$.

To show that $\omega_n(t)$ are subsolutions of $K$, note that $\delta>1$
implies:
\begin{eqnarray}
K\omega_n(t)
&=&z_0(t)+f_0\int_0^t\frac{[z_0(s)+z_1(s)+\dots+z_n(s)]^\delta}{s^\varepsilon}\,ds\nonumber\\
&\ge&z_0(t)+f_0\int_0^t\frac{z_0(s)^\delta+z_1(s)^\delta+\dots+z_n(s)^\delta}{s^\varepsilon}\,ds\nonumber\\
&=&z_0(t)+z_1(t)+z_2(t)+\dots+z_{n+1}(t)=\omega_{n+1}(t)\ge\omega_n(t).\nonumber
\end{eqnarray}



\qed


%ProofProposition2
        {\sc Proof of Proposition \ref{div}.}
(a) Assume that $\varepsilon<1$. Using (\ref{zmm}) we have
        \begin{displaymath}
         z_m(t)\ge\frac{g_0^{\delta^{m}}f_0^{\frac{\delta^m-1}{\delta-1}}
         t^{\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^m+
         \varepsilon-1}{\delta-1}}(\delta-1)^{\sum_{k=1}^m\delta^{m-k}}}
         {[\gamma(\delta-1)-\varepsilon+1]^{\frac{\delta^m-1}{\delta-1}}
         \delta^{\sum_{k=1}^m k\delta^{m-k}}},
	\end{displaymath}
where we have used
$\sum^{m}_{k=1}\delta^{m-k}=\frac{\delta^{m}-1}{\delta-1}$.
Now we use 
$$\sum^{m}_{k=1}k\delta^{m-k}=\frac{(2\delta -1)(\delta^m-1)}{(\delta
         -1)^{2}}-\frac{\delta^m+m-1}{\delta -1}$$
to obtain
\begin{eqnarray*}
z_m(t)&\ge&\left(\frac{[\gamma(\delta-1)-\varepsilon+1]\delta^{\delta'}
t^{\varepsilon-1}}
{f_0(\delta-1)}\right)^{\frac1{\delta-1}}(\delta^m)^{\frac 1{\delta-1}}\\
&&\times \left(\frac{g_0^{\delta-1}f_0
t^{\gamma(\delta-1)-\varepsilon+1}(\delta-1)}
{\delta^{\delta'}[\gamma(\delta-1)-\varepsilon+1]}
\right)^{\frac{\delta^m}{\delta-1}}
\end{eqnarray*}
Note that condition (\ref{hypn}) implies that $t^*\le T$.
Since the inequality $t\ge t^*$ is equivalent to
$$
\frac{g_0^{\delta-1}f_0
t^{\gamma(\delta-1)-\varepsilon+1}(\delta-1)}{\delta^{\delta'}[\gamma(\delta-1)-\varepsilon+1]}\ge1,
$$
we obtain $z_m(t)\ge A(t)\cdot(\delta^m)^{\frac1{\delta-1}}$, where $A(t)$
does not depend on $m$.
Using $\delta>1$ we have that $\delta^m\to\infty$ as $m\to\infty$,
and we conclude that $\sum_{m=0}^{\infty}z_m(t)=\infty$ as
$m\to\infty$ for all $t\in[t^*,T]$.

 (b) Assume that $\varepsilon\ge1$. Since
 $[\gamma(\delta-1)-\varepsilon+1]\delta^k+\varepsilon-1\le\gamma(\delta-1)\delta^k$, we obtain similarly
 as in (a):
$$
 z_m(t)\ge\left(\frac{\gamma\delta^{\delta'}t^{\varepsilon-1}}{f_0}\right)^{\frac1{\delta-1}}
 (\delta^m)^{\frac 1{\delta-1}}\left(\frac{g_0^{\delta-1}f_0
t^{\gamma(\delta-1)-\varepsilon+1}}{\gamma\delta^{\delta'}}\right)^{\frac{\delta^m}{\delta-1}}
$$
for all $t\in[0,T]$ and all $m\in{\bf N}$. The rest of the proof is
analogous to (a).
\qed



\paragraph{Comparison of existence and non-existence regions.}
Let us denote the non-existence set of all $(f_0,g_0)$ satisfying
the conditions of Theorem \ref{non} by ${\cal M}_n$. Of course, we
assume that the constants $\delta$, $\gamma$, and $\varepsilon$ are fixed.
As we already saw, we have existence of solutions of $\omega=K\omega$
for
$(f_0,g_0)\in{\cal M}_s\cup M_m$, see Theorems \ref{schauder} and
\ref{monotone}. We do not know anything about
solvability or nonsolvability of (\ref{fixedpteq}) when
$$
(f_0,g_0)\in (0,\infty)^2\setminus({\cal M}_s\cup{\cal M}_m\cup{\cal M}_n).
$$
If we denote the set of all positive real numbers
$(f_0,g_0)$ for which (\ref{fixedpteq})
is solvable by ${\cal S}$, and the set of all $(f_0,g_0)$ for which the
fixed point equation is not solvable by ${\cal N}$, then we have
${\cal M}_s\cup{\cal M}_m\subseteq{\cal S}$ and ${\cal M}_n\subseteq{\cal N}$. We do not know whether
any of the sets ${\cal S}$ or ${\cal N}$ is closed or open in
$(0,\infty)^2$. However, we conjecture that these two sets are
separated by a curve of the form $f_0=c/g_0^{\delta-1}$, with some
constant $c>0$.

Now we would like to discuss how large is the region in
$(0,\infty)^2$ where we
do not know anything about existence
 or non-existence.
First, we compare the sets ${\cal M}_s$ and ${\cal M}_n$. It will be
convenient to fix $f_0$ and see the quotient of the corresponding
values of $g_n$ and $g_s$ defined as the minimal  possible value of $g_0$ in
(\ref{hypn}) and the maximal possible value of $g_0$
in (\ref{hyps}) respectively. It is not difficult to see that
both for $\varepsilon<1$ and $\varepsilon\ge1$ we obtain the same estimate
\begin{equation}\label{n/s}
\frac{g_n}{g_s}\ge
\delta'\delta^{\frac{\delta'}{\delta-1}}>e^{\delta'-1},
\end{equation}
where $e=2.71828\dots$ The last inequality follows from the fact that
$\delta^{\delta'}(\delta')^{\delta-1}$ is increasing for $\delta>1$ and tends to $e$
as $\delta\to1$.
To show the first inequality in (\ref{n/s}), let $\varepsilon<1$.
Then
$$
\left(\frac{g_n}{g_s}\right)^{\delta-1}=
\frac{\gamma(\delta-1)-\varepsilon+1}{\gamma\delta-\varepsilon+1}\,\delta^{\delta'}(\delta')^\delta,
$$
and the infimum 
of the fraction on the right-hand side over $\varepsilon<1$
is equal to $\frac{\delta-1}\delta$.
Similarly for $\varepsilon\ge1$.

Now let us compare the sets ${\cal M}_m$ and ${\cal M}_n$.
Let us fix $f_0>0$ again and denote the right-hand side
(\ref{sourcem}) by $g_m$.
We obtain
\begin{equation}
\frac{g_n}{g_m}>(e\,\delta^{\delta'})^{\delta'-1}.
\end{equation}
Indeed, let $\varepsilon<1$. Using $e^x\ge ex$ with
$x=\frac{\delta-1}{1-\varepsilon/\delta}f_0T^{1-\varepsilon/\delta}$ we obtain
$$
\left(\frac{g_n}{g_m}\right)^{\delta-1}\ge\frac{\gamma(\delta-1)-\varepsilon+1}
{1-\frac\varepsilon\delta}\,e\,\delta^{\delta'}.
$$
The desired inequality follows using $\gamma>\varepsilon/\delta$. Similarly we can
prove that  (\ref{n/s}) holds also for $\varepsilon\ge1$.

%%%%%%%%%%%%%%%%

\subsection{Qualitative properties of solutions}
\label{qual1}

The following result will be important in obtaining regularity
of solutions of (\ref{pde}), see Proposition~\ref{reg-pde} below.

\begin{prop}\label{reg-odj} Assume that $\delta>\frac{\varepsilon-1}\gamma+1$, $\gamma>0$,
$\varepsilon\in{\mathbb R}$, and let $\omega\in D(K)$ be such that $\omega=K\omega$, where
$K$ is defined (\ref{K}) and $D(K)$ by (\ref{DK}) or (\ref{DKm}).
 Then
\begin{equation}
\lim_{t\to0}\frac{\omega(t)}{t^\gamma}=\lim_{t\to0}\frac{\omega'(t)}{\gamma\,
t^{\gamma-1}}=g_0.
\end{equation}
We have $\omega\in C^{\infty}((0,T])$, and if $\gamma\ge1$ then also $\omega\in
C^1([0,T])$.
\end{prop}

\noindent{\sc Proof.}  Due to L'Hospital's rule it suffices to prove only the
second equality. From (\ref{ode}) we have
$$
\frac{\omega'(t)}{\gamma\, t^{\gamma-1}}=g_0+\frac{\omega(t)^\delta}{t^{\varepsilon+\gamma-1}}.
$$
Since $0\le\omega(t)\le Mt^\gamma$ for some $M>0$, the second term on the
right-hand side tends to zero as
$t\to0$. The fact that $\omega\in C^{\infty}((0,T])$ follows easily from
$\omega=K\omega$. The continuity of $\omega'(t)$ at $t=0$ for $\gamma\ge1$
follows immediately
from (\ref{ode}).
\qed
\medskip


Now we give a partial answer to the question of
 continuous dependence of solutions of
(\ref{ode}) on $(f_0,g_0)\in {\cal M}_s\cup {\cal M}_m$, where ${\cal M}_s$ is defined by
(\ref{hyps}) and ${\cal M}_m$ by (\ref{sourcem}). Recall that ${\cal M}_b\subset{\cal M}_s$.

\begin{prop}\label{cont} Assume that $\delta>\frac{\varepsilon-1}\gamma+1$,
$\delta>0$, $\gamma>0$, $\varepsilon\in{\mathbb R}$.
Assume that $(f_1,g_1)\in{\cal M}_b$, and
let $\omega_1(t)$ be  the unique solution
obtained via Theorem~\ref{banach}.
Let $I(f_1)$ and $I(g_1)$ be closed neighbourhoods of $f_1$
and $g_1$ respectively such that $I(f_1)\times I(g_1)\subset {\cal M}_b$,
and let $f_0=\max I(f_1)$,
$A:=[1-c(f_0)]^{-1}$,
where $c(f_0)$ is defined by (\ref{f00}).
Then for all $f_2\in I(f_1)$, $g_2\in
I(g_1)$ we have
\begin{equation}
\|\omega_1-\omega_2\|_{\gamma}\le A[\,|g_1-g_2|+|f_1-f_2|B(\omega_1)\,],
\end{equation}
with
$$
B(\omega_1)=\sup_{t\in(0,T]}\frac1{t^\gamma}\int_0^T\frac{\omega_1(s)^\delta}{s^\varepsilon}\,ds.
$$
In particular, if $f_2\to f_1$ and $g_2\to g_1$, then
$\omega_2\to\omega_1$ uniformly on $[0,T]$.
\end{prop}

\noindent{\sc Proof.} Since $(f_2,g_2)$ is contained in a convex, open neighbourhood
of $(f_1,g_1)$ in ${\cal M}_b$,
 there exists $M>0$ such that $(f_i,g_i)\in Q_M$, see (\ref{Mb}).
 Therefore the corresponding
operators $K_i$, $i=1,2$
 have a common domain $D(K)$ defined
by (\ref{DK}) with the same $M>0$. Since $\omega_i =K_i\omega_i$, we have for any
$t\in[0,T]$ (see the proof of Theorem \ref{banach}):
\begin{eqnarray}\label{nej}
\frac1{t^\gamma}|\omega_1(t)-\omega_2(t)|&\le&\frac1{t^\gamma}|K_1\omega_1-K_2\omega_1|+
\|K_2\omega_1-K_2\omega_2\|_\gamma
\nonumber\\
&\le&|g_1-g_2|+|f_1-f_2|B(\omega_1)+c(f_2)\|\omega_1-\omega_2\|_\gamma,\\
&\le&|g_1-g_2|+|f_1-f_2|B(\omega_1)+c(f_0)\|\omega_1-\omega_2\|_\gamma,\nonumber
\end{eqnarray}
Taking supremum over $t\in(0,T]$ in (\ref{nej}) and using
$c(f_0)<1$ we obtain the claim.
\qed

Note that we have established continuous dependence of solutions
only in ${\cal M}_b$, i.e.\ in the
proper subset of ${\cal M}_s$. We do not know anything about continuous
dependence of solutions of (\ref{ode}) with respect to $(f_0,g_0)\in
({\cal M}_s\cup{\cal M}_m)\setminus{\cal M}_b$.

Also note that for any given $(f_0,g_0)\in {\cal M}_s$ there exist the minimal
value of $M>0$
 such that $(f_0,g_0)\in T_M$, see (\ref{Ms}). This value, that
 we denote
by $M_e$, appears in an a priori bound of $\omega(t)$.

 \begin{prop}\label{MeMu} Assume that $\delta>\frac{\varepsilon-1}\gamma+1$,
$\delta>0$, $\gamma>0$, and $\varepsilon\in{\mathbb R}$.  Let $(f_0,g_0)\in{\cal M}_s$, see (\ref{Ms}), and
 let $\omega$ be the solution of (\ref{K}) from Theorem \ref{schauder}.
 Assume that $M_e$
 is the smaller of two positive
 solutions of equation (\ref{hypsM}).

(a) We have the following estimate:
 \begin{equation}
 0\le\omega(t)\le M_e t^\gamma.
 \end{equation}
Furthermore, this solution of (\ref{ode}) is unique in the set
defined by (\ref{DKm}).

(b) If $(f_0,g_0)\in {\cal M}_s$, see (\ref{Ms}), then
\begin{equation}\label{bezf}
|\omega(b)-\omega(a)|\le g_0|b^\gamma-a^\gamma|+f_0
M_e^\delta\frac{|b^{\gamma\delta-\varepsilon+1}-a^{\gamma\delta-\varepsilon+1}|}{\gamma\delta-\varepsilon+1},\quad\forall
a,b\in(0,T).
\end{equation}
In particular,
\begin{equation}\label{bezf2}
|\omega'(t)|\le
g_0\gamma t^{\gamma-1}+f_0
M_e^\delta t^{\gamma\delta-\varepsilon}.
\end{equation}
\end{prop}

\noindent{\sc Proof.} (a) Note that $\omega\in D(K)$, with $D(K)$ as in
(\ref{DK}) and $M=M_e$. The uniqueness claim follows from
Lemma~\ref{unique}.

(b) We have that $0\le\omega(t)\le M_et^\gamma$, and there
holds (\ref{source}). Hence, similarly as in the proof of
equicontinuity in Lemma \ref{compact} we get that for all $a$,
$b\in[0,T]$, $a<b$:
\begin{equation}
|\omega(b)-\omega(a)|=|K\omega(b)-K\omega(a)|\le g_0|b^\gamma-a^\gamma|+\frac{f_0
M_e^\delta}{\gamma\delta-\varepsilon+1}[\,b^{\gamma\delta-\varepsilon+1}-a^{\gamma\delta-\varepsilon+1}\,]
\end{equation}
Relation (\ref{bezf2}) follows if we divide (\ref{bezf}) by
$|b-a|$ and let $b\to a$.
\qed

It is possible to effectively compute $M_e$.
For example, if $\delta=2$ it is easy to see that
\begin{equation}\label{MeMu2}
M_e=\frac{1-\sqrt{1-4a g_0}}{2a},
\quad a=\frac{f_0
T^{\gamma-\varepsilon+1}}{2\gamma-\varepsilon+1}.
\end{equation}
Note that in the case of monotone iterations method (see Theorem
\ref{monotone}), that is, when $(f_0,g_0)\in {\cal M}_m$,
we have uniqueness of the corresponding solution
in a much larger domain, which
is defined by (\ref{DKm}).

%%%%%%%%%%%%%%%%%%%

\section{Quasilinear PDE}
\label{quasi}

\subsection{Existence and non-existence of solutions}

The main results of this paper are Theorem~\ref{main}
and Theorem~\ref{uniqqque} below.
Recall that by $\omega$-solutions of (\ref{pde}) we mean strong
solutions that can be obtained via ODE (\ref{ode}), as described
in Lemma~\ref{conn}.

\begin{theorem}\label{main} Assume that $1<p<\infty$,
$m>\max\{-p,-N\}$,
and let $\tilde f_0$ and $\tilde g_0$ be positive real numbers.

(a) If
\begin{equation}\label{hypspde}
\tilde f_0\le\left(m+1+\frac N{p'}\right)
\left(\frac{m+N}{pR^{m+p}\tilde g_0}\right)^{p'-1},
\end{equation}
then there exists  $\omega$-solution of (\ref{pde})
which can be obtained
constructively using the method of monotone iterations, see
the remark after the proof of Theorem \ref{monotone}, and Lemma
\ref{conn}.
If we have strict inequality in (\ref{hypspde}) then
the same sequence of monotone iterations
can be obtained also by
contraction method via Theorem \ref{banach} and Lemma \ref{conn}.
Any $\omega$-solution of
(\ref{pde}) is unique
in the set
\begin{equation}\label{uniqque}
\{w\in C^2(B\setminus\{0\})\cap C(\overline B)\,:\,\exists M_w>0,\,\,|\nabla
w(x)|\le M_w
|x|^{\frac{p'}p(m+1)}\}.
\end{equation}
and satisfies the following estimate:
\begin{equation}\label{estimatex}
|\nabla v(x)|\le
NC_N^{\frac{(m+p)p'}{Np}}M_e^{p'/p}|x|^{\frac{p'}p(m+1)},
\end{equation}
where $M_e=M_e(f_0,g_0)$ is defined in Proposition \ref{MeMu}
for $f_0$ and $g_0$ as in (\ref{veza}).
In particular, in the case of $p=2$ and $m=0$  we have that
\begin{equation}
|\nabla v(x)|\le |x|\frac{N+2}{2\tilde f_0R^2}\left(1-
\sqrt{1-\frac{4\tilde f_0\tilde g_0R^2}{N(N+2)}}\,\right).
\end{equation}

(b) If $m\ge-1$ and
\begin{equation}\label{hypm}
\displaystyle\tilde g_0\le N^{p-1}C_N^{\frac{m+p}N}(m+N)|B|^{-\frac{m+1}N}
e^{-N|B|^{1/N}\tilde f_0},\quad |B|=C_NR^N,
\end{equation}
then there exists  $\omega$-solution of (\ref{pde})
that can be obtained constructively
using the method of monotone iterations via Theorem
\ref{monotone} and Lemma \ref{conn}, and which is unique in
the set (\ref{uniqque}).
If  $(\tilde f_0,\tilde g_0)$ satisfies also the condition in~(a),
then the solution in~(b) coincides with the one in~(a).

(c) If
\begin{equation}\label{nonpde}
\tilde f_0
\ge
\left\{
\begin{array}{ll}
\displaystyle (m+p)(p')^p\left(\frac{m+N}{R^{m+p}\tilde
g_0}\right)^{p'-1}& \mbox{ for $p>N$,}\\
\displaystyle (m+N)(p')^p\left(\frac{m+N}
{R^{m+p}\tilde g_0}\right)^{p'-1}& \mbox{ for $p\le N$,}
\end{array}
\right.
\end{equation}
then (\ref{pde}) has neither strong nor weak solutions.
\end{theorem}

\noindent{\sc Proof.} It suffices
to use Lemma \ref{conn} together with:
(a)~Theorem \ref{schauder}, Theorem \ref{banach} and
Proposition \ref{MeMu},  (b)~Theorem
\ref{monotone}. (c)~To show non-existence, we proceed by
contradiction. Assume that $v$ is a strong or weak solution of (\ref{pde}).
Using Lemma~\ref{conno} or Theorem~\ref{thw2}
respectively, in both cases we obtain a solution $\omega\in D^+$ of
(\ref{odde}). This contradicts Theorem~\ref{non}.

Coincidence of
solutions  in~(a) and~(b) follows easily from
the fact that the uniqueness domain in~(b) equals the
uniqueness domain in~(a).
\qed

In Theorem~\ref{uniqqque} we will show that $\omega$-solutions, whose
existence we have proved in the above theorem, see (a) and (b), are
in fact unique weak solutions of
(\ref{pde}).

It is natural to define the sets
$\tilde{\cal M}_s$,
$\tilde{\cal M}_m$,
and $\tilde{\cal M}_n$ of all $(\tilde f_0,\tilde g_0)\in(0,\infty)^2$ satisfying
conditions in~(a), (b), and~(c)  respectively. If
$$
(\tilde f_0,\tilde g_0)\in(0,\infty)^2\setminus(
\tilde{\cal M}_s\cup
\tilde{\cal M}_m\cup
\tilde{\cal M}_n),
$$
we do not know anything about
existence or non-existence of weak and strong solutions of (\ref{pde}).
Also, let us denote by $\tilde{\cal S}$ and $\tilde{\cal N}$ the existence and
non-existence sets of all $(\tilde f_0,\tilde g_0)$ in
$(0,\infty)^2$. We do not know anything about geometrical
properties of these sets, except that
$$
\tilde {\cal S}\supseteq \tilde {\cal M}_s\cup \tilde {\cal M}_m,\quad \tilde
{\cal N}\supseteq\tilde{\cal M}_n.
$$

%%%%%%%%%%%%%%


\subsection{Qualitative properties of solutions}
\label{qual2}

First we study the behaviour of the gradient of $\omega$-solutions
of (\ref{pde}) near the origin $x=0$
and near the boundary of ball $B$.

\begin{lemma}\label{urM} Let $m>\max\{-p,-N\}$, and let $v(x)$ be $\omega$-solution
of (\ref{pde}). Let us  define $u(r)=v(x)$, $r=|x|$. Then

(a) $v\in C^{\infty}(\overline B\setminus\{0\})\cap C(\overline B)$;

(b)
\begin{eqnarray}
\lim_{r\to0}\frac{u'(r)}{r^{\frac{p'}p(m+1)}}&=&-\left(\frac{\tilde
g_0}{m+N}\right)^{p'/p}\label{regul}\\
\lim_{r\to0}\frac{u''(r)}{r^{\frac{p'}p(m-p+2)}}&=&
-\frac{m+1}{p-1}\left(\frac{\tilde
g_0}{m+N}\right)^{p'/p},\label{regull}
\end{eqnarray}

(c)  if  $(\tilde f_0,\tilde g_0)\in\tilde{\cal M}_s$, see
(\ref{hypspde}),  then
\begin{equation}\label{u'Rg}
|u'(R)|\le
D_1R^{\frac{p'}p(m+1)}+D_2 R^{\frac{2p'}p(m+1)+1},
\end{equation}
where
$$
D_1=\frac{\tilde g_0 N^2M_e^{p'-2}}{m+N}C_N^{\frac{m(p'-2)+p'-p}N},\;
D_2=\frac{\tilde f_0 p
N^2 M_e^{2(p'-1)}}
{p'(2m+N+1)+p} C_N^{\frac{p'}p\frac{m+p}N+\frac{p'(m+1)-m}N}.
$$
Here $M_e=M_e(f_0,g_0)$ is defined in Proposition \ref{MeMu}
with $f_0$, $g_0$ from (\ref{veza}).
\end{lemma}

\noindent{\sc Proof.} (a) This follows easily from (\ref{vV}), (\ref{V}) and
Proposition~\ref{reg-odj}.

(b) It suffices to use (\ref{osn2}). The
first relation is immediate, while the second relation follows after
differentiating (\ref{osn2}) and using Proposition \ref{reg-odj}
with $t=C_Nr^N$. Indeed, we have that $\frac{\omega(t)}{t^\gamma}\to g_0$,
$\frac{\omega'(t)}{\gamma\,t^{\gamma-1}}\to
g_0$ as $t\to0$.
Denoting $c=-
N{C_N^{\frac{p'}p(\frac pN-1)}}$ and $\gamma=1+\frac mN$
we obtain after an easy
computation that
\begin{eqnarray}
\frac{u''(r)}{c\,r^{\frac{p'}p(m-p+2)}}&=&
-(N-1)C_N^{\frac{\gamma
p'}p}\frac{p'}p\left[\frac{\omega(C_Nr^N)}{(C_Nr^N)^\gamma}
\right]^{p'/p}+
\nonumber\\
&\phantom{=}&+\gamma NC_N^{\frac {p'\gamma}p}\frac{p'}p
\left[\frac{\omega(C_Nr^N)}{(C_Nr^N)^\gamma}\right]^{\frac{p'}p-1}
\frac{\omega'(C_Nr^N)}{\gamma(C_Nr^N)^{\gamma-1}}\nonumber\\
&\to&-(N-1)C_N^{\frac{\gamma
p'}p}\frac{p'}pg_0^{p'/p}+\gamma NC_N^{\frac {p'\gamma}p}\frac{p'}p
g_0^{\frac{p'}p-1}g_0\nonumber\\
&=&\frac{p'}pg_0^{p'/p}C_N^{\gamma p'/p}(\gamma N-N+1)
\quad\mbox{as $r\to0$.}
\nonumber
\end{eqnarray}
Now we use (\ref{veza})  to obtain the
desired result.

(c) To prove (\ref{u'Rg}) we first use
(\ref{osn2}):
$$
r^{\frac{p'}p(N-1)}|u'(r)|=|c|\cdot\omega(C_Nr^N)^{p'/p}.
$$
Now we differentiate this equality with respect to $r$:
$$
\od{}r(r^{\frac{p'}p(N-1)}|u'(r)|)=|c|\frac{p'}p\omega(C_Nr^N)^{\frac{p'}p-1}
\omega'(C_Nr^N)\cdot C_N\,N\,r^{N-1},
$$
and then use estimates $0\le \omega(t)\le Mt^\gamma$ and
(\ref{bezf2}).
 The desired inequality follows after a short
computation upon
integration over $(0,R)$:
$$
R^{\frac{p'}p(N-1)}|u'(R)|-\lim_{r\to0}r^{\frac{p'}p(N-1)}|u'(r)|=
\int_0^R[E_1r^{\frac{p'}p(m+N)-1}+E_2r^{\frac{p'}p(2m+N+1)}]dr,
$$
where $E_1$, $E_2$ are positive constants depending on $m$,
$p$, $N$, $\tilde g_0$, and $\tilde f_0$.
Note that the function under the integral sign is
integrable
since $m>\max\{-N,-p\}$ implies that the exponents at $r$ are
$>-1$.
We also need (\ref{regul}) to obtain that
$$
\lim_{r\to0}r^{\frac{p'}p(N-1)}|u'(r)|=|A|\lim_{r\to0}r^{\frac{p'}p(m+N)}=0,
$$
where $A$ is the right-hand side of (\ref{regul}).
\qed


\medskip
A priori bound (\ref{u'Rg}) is a refinement of (\ref{estimatex}) for $|x|=R$.
Relations (\ref{regul}) and (\ref{regull}) immediately imply
the following qualitative properties
of the gradient of $\omega$-solutions of (\ref{pde}) at $x=0$.

\begin{prop}\label{reg-pde} Assume that $m>\max\{-p,-N\}$, and let $v(x)$ be
$\omega$-solution of (\ref{pde}).  Then
 we have the following
regularity results at $x=0$:

(a) If $m<-1$ then $\lim_{r\to0}u'(r)=-\infty$. In particular,
$v\notin C^1(\overline B)$.

(b) If $m=-1$ then
\begin{equation}
\lim_{r\to0}u'(r)=-\left(\frac{\tilde g_0}{m+N}\right)^{p'/p}
\end{equation}
As in case (a), we have $v\notin C^1(\overline B)$.

(c) If $-1<m<p-2$, then
\begin{equation}
\lim_{r\to0}u'(r)=0,\quad
\lim_{r\to0}u''(r)=-\infty.
\end{equation}
In particular, $v\in C^1(\overline B)$ and $v\notin C^2(B)$.

(d) If $m\ge p-2$, then $\lim_{r\to0}u'(r)=0$ and
\begin{equation}
\lim_{r\to0}u''(r)=\left\{
\begin{array}{ll}-\frac{m+1}{p-1}\left(\frac{\tilde
g_0}{m+N}\right)^{p'/p}& \mbox{for $m=p-2$,}\\
0& \mbox{for $m>p-2$.}
\end{array}\right.
\end{equation}
In particular, $v$ is classical solution, $v\in C^2(\overline B)$.
\end{prop}

The above proposition shows that we have precise information on the
gradient of $\omega$-solutions at $x=0$. With larger values of $m$ we
can have even more regularity of $\omega$-solutions, and it is
possible to study H\"older continuity as well, see
\cite{dz-crit}. On the other hand, we are not able to obtain
precise information about $v(0)$. We can obtain only upper and
lower estimates for $v(0)$.

\begin{prop}\label{estimates} (a priori estimate of $v(0)$)
 (a) Let $m>\max\{-p,-N\}$ and let $v(x)$ be $\omega$-solution
of (\ref{pde}) corresponding to $(\tilde f_0,\tilde g_0)\in\tilde {\cal M}_s$,
see Theorem \ref{schauder}, and let $M_e=M_e(f_0,g_0)$ be defined as
in Proposition \ref{MeMu},
where $f_0,g_0$ are given by (\ref{veza}).
Then
\begin{equation}
v(0)\le N\frac{p-1}{m+p}\cdot C_N^{\frac{m+p}N\cdot\frac{p'}p}
R^{\frac{p'}p(m+p)}M_e^{p'/p}.
\end{equation}
In particular, for $p=2$ and $m=0$ we have
\begin{equation}
v(0)\le\frac{N+2}{4\tilde f_0}\left(1-\sqrt{1-\frac{4\tilde
f_0\tilde g_0 R^2}{N(N+2)}}\,\right)
\end{equation}

(b) For any weak solution $v(x)$ of (\ref{pde}) we have the following lower bound:
\begin{equation}\label{oscC}
v(0)\ge
\left\{
\begin{array}{ll}
\displaystyle\frac1{{(2p)}^{p'}}
\left(\frac{R^{m+p}\tilde g_0}
{2^N-1}\right)^{p'-1}& \mbox{for $m<0$,}\\
\displaystyle\left(\frac{c(m,p,N)R^{m+p}\tilde
g_0}{p^p}\right)^{p'-1}& \mbox{for $m\ge0$,}
\end{array}
\right.
\end{equation}
where
\begin{equation}
c(m,p,N)=\sup_{t\in(0,\frac12)}\frac{t^p[(1-t)^N-t^N](1-t)^m}
{1+t^N-(1-t)^N}.
\end{equation}
If $m>\max\{-p,-N\}$  then the above estimate holds also
for any $\omega$-solution
of (\ref{pde}).

In particular, when $p=2$, $N=2$, $m\ge0$, we obtain the following
lower bound for weak solutions of (\ref{pde}):
\begin{equation}
v(0)\ge\frac14c(m)R^{m+2}\tilde g_0,
\end{equation}
where
$$
c(m)=\frac12 t(1-2t)(1-t)^m,\quad
t=\frac{m+5-\sqrt{m^2+2m+9}}{4(m+2)}.
$$
If in addition to this we assume that $m=0$,
we obtain $v(0)\ge\frac1{64}R^2\tilde g_0$.


(c) If $m\ge-1$ and $(\tilde f_0,\tilde g_0)\in\tilde
{\cal M}_m$, see (\ref{hypm}),
then for the corresponding $\omega$-solution
$v(x)$ of (\ref{pde}) we have
\begin{eqnarray}
&\displaystyle c(p)\frac{p-1}{m+p}\left[
\left(\frac{\tilde
g_0}{m+N}\right)^{p'/p}R^{p'(m+1)-m}+\right.\nonumber&\\
&\displaystyle\left.+\left(\frac{\tilde f_0\tilde
g_0^{p'}}{[p'(m+1)+N](m+N)^{p'}}\right)^{p'/p}
(p')^{-1}R^{p'[p'(m+1)-m]}
\right]\le\label{ocj-mon}&
\\
&\displaystyle\le v(0)\le
 \frac{p-1}{m+p}\left(\frac{\tilde g_0\cdot e^{\tilde
f_0N|B|^{1/N}} R^{m+p}} {m+N}\right)^{p'/p},\nonumber&
\end{eqnarray}
where $c(p)=1$ for $p\in(1,2)$ and $c(p)=2^{p'-2}$ for $p\ge2$.
The same lower bound holds also for $(\tilde f_0,\tilde
g_0)\in\tilde{\cal M}_s$, see (\ref{hypspde}).
\end{prop}

\noindent{\sc Proof.}
(a) We obtain the upper bound using (\ref{osn2})
and $v(0)\le\int_0^R|u'(r)|\,dr$.

(b) This lower bound is obtained using
 lower oscillation estimate for general quasilinear elliptic
 problems studied in \cite{kpz}.
In the case of
$m<0$ the estimate follows immediately from
\cite[Corollary~12]{kpz}.
If $m\ge0$, we use the following version of
oscillation estimate, see \cite[Theorem~9]{kpz}:
\begin{equation}
\mathop{\rm osc}_{A_r} v\ge\frac1{p^{p'}}\left(\frac{r^p|A|}{|A_r\setminus
A|}\right)^{p'/p}\mathop{\rm ess\,inf}_{x\in A}(\tilde g_0|x|^m)^{p'/p},
\end{equation}
for any open subset $A\subset B$ and $r>0$ such that $A_r\subset B$,
where $A_r$ denotes open $r$-neighbourhood of $A$.
Note that $\mathop{\rm osc}_B v=v(0)$.
We consider the family of sub-rings $A$ of $B$ such that
$A_r=B\setminus\{0\}$,
$r\in(0,\frac R2)$. Then substituting $t=\frac rR$ we obtain
\begin{eqnarray}
v(0)&\ge&\frac{\tilde g_0^{p'/p}}{p^{p'}}\sup_{r\in(0,\frac R2)}
\left(\frac{r^p[(R-r)^N-r^N]}{R^N+r^N-(R-r)^N}\right)^{p'/p}(R-r)^{mp'/p}\nonumber\\
&=&\frac{\tilde g_0^{p'/p}}{p^{p'}}\left(\sup_{r\in(0,\frac R2)}
\frac{r^p[(R-r)^N-r^N](R-r)^m}{R^N+r^N-(R-r)^N}\right)^{p'/p}\\
&=&\frac{\tilde g_0^{p'/p}}{p^{p'}} R^{\frac{p'}p(m+p)}
\left(\sup_{t\in(0,\frac12)}\frac{t^p[(1-t)^N-t^N](1-t)^m}
{1+t^N-(1-t)^N}\right)^{p'/p}.\nonumber
\end{eqnarray}
It will be shown in Proposition~\ref{unicque} that for
$m>\max\{-p,-N\}$ any $\omega$-solution is also weak
solution.

(c) The claim follows from estimate (\ref{o<v})
in Theorem \ref{monotone} and from Lemma \ref{conn}, similarly as
in (a); see also the remark after the proof of Theorem
\ref{monotone}. We use the fact that $(a+b)^q\ge d(q)(a^q+b^q)$, where
$d(q)=2^{q-1}$ for $q\in(0,1)$, $d(q)=1$ for $q\ge1$, $a\ge0$,
$b\ge0$, with $q=p'/p$.
\qed

Note that in particular, under the assumptions of (a), we have
that for any $\omega$-solution $v(x)$ of (\ref{pde}) we have
\begin{equation}
\mbox{if $R\to0$ then $v(0)\to0$.}
\end{equation}
Property (b) implies that for any weak solution $v(x)$ we have
\begin{equation}
\mbox{if $R\to\infty$ or $\tilde g_0\to\infty$, then $v(0)\to\infty$.}
\end{equation}
Under the assumptions of (c) we have that all these
implications hold for $\omega$-solutions. Furthermore, from (\ref{ocj-mon}) we also see that
\begin{equation}
\mbox{if $\tilde f_0\tilde g_0^{p'}\to\infty$ then $ v(0)\to\infty$}.
\end{equation}
It is possible to obtain even better constant than $c(m,p,N)$
in (\ref{oscC}), if in the proof
we consider the family of sub-annuli $A$ such
that $A_r\subset B\setminus\{0\}$, and not only
$A_r=B\setminus\{0\}$.
\medskip

In formulating our problem (\ref{pde}) we imposed the condition that
$v(x)$ be decreasing.
Let us consider the corresponding  problem without this
condition:
\begin{eqnarray}
&-\Delta_p v=\tilde g_0|x|^m+\tilde f_0|\nabla v|^p,& \nonumber\\
&v|_{\partial B}=0\,,\quad v\in C^2(B\setminus\{0\})\cap C(\overline B),&
\label{bezdecr}\\
&v(x) \mbox{ spherically symmetric.}& \nonumber
\end{eqnarray}
The following proposition shows that if
$v(x)$ is a strong solution of (\ref{bezdecr}) which belongs to
$C^1(B)$, or $p>2$, then $u(r)$ is necessarily decreasing, where
$u(r)=v(x)$, $r=|x|$.

\begin{prop}\label{decr} Let $\tilde f_0$ and $\tilde g_0$ be
positive real numbers. Let $v(x)$ be a strong solution of
(\ref{pde}).

(a) Then
$u|_{(0,R)}$
can have at most one local maximum $r_0$,
and $u(r)$ is decreasing on $(r_0,R)$.

(b) If  $v\in C^1(B)$, then $u(r)$ is decreasing on $[0,R]$.

(c) If $p>2$ then $u(r)$ is decreasing on $[0,R]$.
\end{prop}

\noindent{\sc Proof.} (a) Let us define $V(s)$, $s\in(0,|B|)$ by
$V(C_Nr^N)=u(r)$.
It suffices to show that if $r_0\in[0,R]$ is such
that
$V'(s_0)=0$, $s_0=C_Nr_0^N$, then $V'(s)<0$ for
$s=C_Nr^N\in(s_0,C_NR^N)$.
Let $x\in B$ be such that $s=C_N|x|^N$ for some given $s>s_0$.
We have
\begin{eqnarray}
-N^pC_N^{\frac pN}\od{}s\left(s^{p(1-\frac1N)}\left|\od Vs\right|^{p-2}\od Vs\right)
&=&-\Delta_pv=\tilde g_0|x|^m+\tilde f_0|\nabla v|^p\nonumber\\
&\ge&\tilde g_0 \left(\frac s{C_N}\right)^{\frac
mN}.\label{nejVs}
%+\tilde f_0 C_N^{1/N}N^ps^{p'(1-1/N)}\left|\od Vs\right|^p.
\end{eqnarray}
Integrating from $s_0$ to $s$ we obtain
$$
-s^{p(1-\frac1N)}\left|\od Vs(s)\right|^{p-2}\od Vs(s)
+s_0^{p(1-\frac1N)}\left|\od Vs(s_0)\right|^{p-2}\od Vs(s_0)
>0.
$$
Since $\od V s(s_0)=0$ we arrive to  $\od Vs<0$,
which proves that $\od ur<0$ for all $r\in (r_0,R)$, see
(\ref{vV}).

(b) If $v\in C^1(B)$, then $u'(0)=0$, and we can proceed as in
(a) with
$r_0=0$.

(c) Assume that $u(r)$ is not decreasing. Then there exists
$r_0\in(0,R)$ such that $u'(r_0)=0$. From
(\ref{nejVs})  we obtain
$$
p(1-\frac1N)s^{p(1-\frac1N)-1}|V'(s)|^{p-2}V'(s)+
(p-1)s^{p(1-\frac1N)}|V'(s)|^{p-2}V''(s)\le -c\cdot s^{m/N},
$$
where $c>0$. Substituting $s=s_0$ we obtain a contradiction:
$0\le-c\cdot s_0^{m/N}$.

Assume that $u$ is decreasing on $(0,a)$, but is not decreasing on
$[a,R]$. Then there exists a point
$r_0\in[r_0,R)$ such that $u'(r_0)=0$, which implies a
contradiction in the same way as above.
\qed
\medskip

In (\ref{u'Rg})
we obtained an upper estimate of $|u'(R)|$, that
is, of the outward normal derivative
on the boundary of
$B$ for any $\omega$-solution $v(x)$ of (\ref{pde}).
Now we want to obtain the lower bound of $|u'(R)|$ for any solution
of (\ref{pde}).

\begin{prop} Let $m>-N$.
For any strong  solution $v(x)$ of (\ref{pde}) such that
$u'(r)\to0$ as $r\to0$ we have the following lower bound:
\begin{equation}
\displaystyle
|u'(R)|\ge\left(\frac{\tilde g_0 R^{m+1}}{m+N}\right)^{p'-1}.
\end{equation}
In particular, if $m>-1$ this estimate holds for any $\omega$-solution
of (\ref{pde}).
\end{prop}
\noindent{\sc Proof.}
Let $s=C_Nr^N$, and let us integrate (\ref{nejVs}) from
$0$ to $|B|$. We obtain
$$
N^pC^{\frac pN}_N|B|^{p(1-\frac1N)}|V'(|B|)|^{p-1}\ge\frac{N\tilde
g_0|B|^{\frac{m+N}N}}{C_N^{m/N}(m+N)}.
$$
On the other hand, after  differentiating
$V(C_Nr^N)=u(r)$ with respect to $r$, we get
$V'(|B|)=\frac{u'(R)}{NC_NR^{N-1}}$, and the result follows
easily. The claim for $\omega$-solutions follows from
Proposition~\ref{reg-pde}.
\qed

We can also state a continuous dependence result for $\omega$-solutions of
(\ref{pde}), which follows
easily from Theorem \ref{main}(a) and Lemma \ref{conn}.

\begin{prop}\label{contpde} Assume that $1<p<\infty$,
$m>\max\{-p,-N\}$ and
 $(\tilde f_1,\tilde g_1)\in \tilde{\cal M}_b$,
 where $\tilde{\cal M}_b$ is defined by
 (\ref{Mb}).
Let $I(\tilde f_1)$ and $I(\tilde
g_1)$ be closed neighbourhoods  of $\tilde f_1$
and $\tilde g_1$ respectively, $f_0=\max I(f_1)$ and $g_0=\max
I(g_1)$, where $f_1$,  $g_1$ and the corresponding intervals
$I(f_1)$, $I(g_1)$
are defined by
(\ref{veza}),
 $A>0$, $B(\omega_1)$ as in Proposition \ref{cont}.
If $\tilde f_2\in
I(\tilde f_1)$ and  $\tilde g_2\in I(\tilde g_1)$,
then for the corresponding $\omega$-solutions $v_1$, $v_2$ from
Theorem~\ref{main}(a) we have that for all
$r\in[0,R]$:
\begin{eqnarray}
|u_1'(r)-u_2'(r)|&\le &C\, r^{\frac{p'}p(m+1)}
[\,|g_1-g_2|+B(\omega_1)|f_1-f_2|\,], \label{razl-der} \\
|u_1(r)-u_2(r)|&\le& C\,\frac{R^{p'(m+1)-m}-r^{p'(m+1)-m}}
{p'(m+1)-m}  \nonumber\\
&&\times [\,|g_1-g_2|+B(\omega_1)|f_1-f_2|\,]\,, \label{Cty}
\end{eqnarray}
where
\begin{eqnarray}
C=(p'-1)NC_N^{\frac{m+p}{N(p-1)}}M_e^{p'-2}A\,.
\end{eqnarray}
Here $f$'s, $g$'s, $u$'s, and $\omega_1$ are defined analogously as in Lemma
\ref{conn},
and
$
M_e=M_e(f_0,g_0)
$,
see Proposition \ref{MeMu}, with $f_0$ and $g_0$ as in
Proposition~\ref{cont}.
In particular, if $\tilde f_2\to \tilde f_1$ and $\tilde g_2\to
\tilde g_1$, then
$v_2\to v_1$ in $C^1(\overline B)$. If $p=2$, then $v_2\to v_1$ in
$C^2(\overline B)$.
\end{prop}

\noindent{\sc Proof.}
(a)  We have
\begin{equation}\label{razlika}\displaystyle
|u_1'(r)-u_2'(r)|=c r^{-(N-1)\frac{p'}p}|\omega_1(C_Nr^N)^{\frac{p'}p}
-\omega_2(C_Nr^N)^{\frac{p'}p}|,
\end{equation}
where $c=NC_N^{\frac{p'}p(\frac pN-1)}$.
We can use (\ref{ineq}) and
$\omega(t)\le Mt^\gamma$, $t=C_Nr^N$, $\gamma=1+\frac mN$, together with
$$
|\omega_1(t)-\omega_2(t)|\le t^\gamma A[\,|g_1-g_2|+B(\omega_1)|f_1-f_2|\,],
$$
see Proposition \ref{cont}.
(\ref{razlika}).

(b) To prove (\ref{Cty}) note that the zero boundary condition
implies
$$
|u_1(r)-u_2(r)|\le\int_r^R|u_1'(\rho)-u_2'(\rho)|\,d\rho.
$$
It suffices to use (\ref{razl-der}).

(c) The convergence in $C^2(\overline B)$ for $p=2$
follows from standard $L^2$-regularity for
elliptic equations.
\qed

%%%%%%%%%%%%%%%%



\subsection{Uniqueness of weak solutions}

Here we study the problem of existence of unique
 weak solution of (\ref{pde}).

\begin{prop}\label{unicque} Assume that $m>\max\{-p,-N\}$. (a)~Then
any $\omega$-solution of (\ref{pde}) is weak solution, and
conversely, any weak solution of (\ref{pde}) is $\omega$-solution.
(b)~There exists at most one weak solution of (\ref{pde}).
In particular, under conditions~(a) or~(b) of Theorem~\ref{main}
problem (\ref{pde}) possesses the unique weak solution.
\end{prop}

\noindent{\sc Proof.} (a1)
Let us show that the pointwise derivative
$\pd v{x_i}$ of $\omega$-solution $v$ is also the weak derivative.
First, since $v$ is continuous, it is integrable on $B$.
We can write
$$
\int_Bv\pd\varphi{x_i}\,dx=\lim_{\varepsilon\to0}\int_{\Omega_\varepsilon}v\pd\varphi{x_i}\,dx,
$$
where $\varphi\in C_0^{\infty}(\Omega)$.
Using Green's formula we have that
$$
\int_{\Omega_\varepsilon}v\cdot\pd \varphi{x_i}\,dx=-\int_{\Omega_\varepsilon}\pd
v{x_i}\cdot\varphi\,dx+\int_{S_\varepsilon}v\,\varphi\,\nu_i\,dS,
$$
where $\nu$ is the outward unit normal vector at $x$, $|x|=\varepsilon$,
with respect to domain $\Omega_\varepsilon=B\setminus B_\varepsilon(0)$, and $S_\varepsilon$ is
the inner bounding sphere of $\Omega_\varepsilon$ whose radius is $\varepsilon$.
The last integral tends to
zero as $\varepsilon\to0$ since  $v$ is bounded. From this we can easily
see that
$$
\int_Bv\cdot\pd\varphi{x_i}=-\int_B\pd v{x_i}\cdot\varphi\,dx,
$$
i.e.\ the pointwise derivative of $v$ is also the weak derivative of
$v$.

(a2) Let us prove that the $\omega$-solution $v$ of (\ref{pde}) is
also weak solution. Since $v$ is of class $C^{\infty}$ on $\Omega_\varepsilon$, see
Lemma~\ref{urM}(c), we have that it satisfies (\ref{pde})
pointwise on $\Omega_\varepsilon$. This together with Green's formula yields:
\begin{eqnarray}
&\displaystyle\tilde g_0\int_{\Omega_\varepsilon}|x|^m\varphi\,dx
+\tilde f_0\int_{\Omega_\varepsilon}|\nabla
v|^p\varphi\,dx=-\int_{\Omega_\varepsilon}\Delta_pv\,\varphi\,dx=&\nonumber\\
&\displaystyle=\int_{\Omega_\varepsilon}|\nabla v|^{p-2}\nabla
v\cdot\nabla\varphi\,dx-
\int_{S_\varepsilon}\sum_{i=1}^N|\nabla v|^{p-2}\pd
v{x_i}\,\varphi\,\nu_i\, dS.&\nonumber
\end{eqnarray}
To show that the last integral tends to zero
we use the fact that there exists a constant $C>0$
such that
$$
|u'(r)|\le C\cdot
r^{(m+1)\frac{p'}p}
$$
for all $r\in[0,R]$, see Lemma~\ref{urM}(b).
Therefore the last integral does not exceed
\begin{equation}
C\int_{S_\varepsilon}|\nabla v|^{p-1}dS\le C|u'(\varepsilon)|^{p-1}\cdot\varepsilon^{N-1}\le
C\cdot \varepsilon^{m+N}\to0\quad\mbox{as $\varepsilon\to0$,}
\end{equation}
where $C>0$ is a generic constant.
Passing to the limit in the above integral equality we obtain
that $-\Delta_p v=\tilde g_0|x|^m+\tilde f_0|\nabla v|^p$ in the weak sense.

It remains to check that $v\in\wo1pB$, that is,
$$
\int_B|\nabla v|^pdx\le C\int _0^Rr^{(m+1)p'+N-1}dr<\infty.
$$
This is equivalent to $(m+1)p'+N>0$, and this inequality follows easily from
$m>\max\{-p,-N\}$.

(b) Since any weak solution is $\omega$-solution,
uniqueness of weak solutions follows from Lemma~\ref{unique}.
\qed

\begin{theorem}\label{uniqqque} Assume that $1<p<\infty$, $m>\max\{-p,-N\}$, and
$$
(\tilde f_0,\tilde g_0)\in \tilde{\cal M}_s\cup\tilde{\cal M}_n,
$$
where $\tilde {\cal M}_s$ and $\tilde {\cal M}_n$ are subsets of $(0,\infty)^2$
defined by (\ref{hypspde}) and (\ref{hypm}) respectively.
Then there exists a unique weak solution $v\in\wo1pB\cap
L^{\infty}(B)$ of (\ref{pde}). Furthermore, we have $v\in C^{\infty}(\overline
B\setminus\{0\})\cap C(\overline B)$, and $v$ is $\omega$-solution.
It has all qualitative properties described in Section~\ref{qual2}.
\end{theorem}

\noindent{\sc Proof.} By Proposition~\ref{unicque} we know that for
$m>\max\{-p,-N\}$ any $\omega$-solution is also weak solution of
(\ref{pde}). Therefore it suffices to show that there exists a
unique $\omega$-solution.
The existence of $\omega$-solutions has been proved
in Theorem~\ref{main}(a) and (b).
Now by Theorem~\ref{thw2} we know that weak solutions of
(\ref{pde}) are in fact $\omega$-solutions.
Assume that there exist two different weak solutions $v_1$ and
$v_2$ of (\ref{pde}). Then this implies the existence of two
different functions $\omega_1$ and $\omega_2$ both satisfying equation
$\omega=K\omega$, where $K$ is defined by (\ref{K}) on domain
(\ref{DKm}). The fact that $\omega_1\ne\omega_1$ follows easily from
(\ref{vV}). But this contradicts Proposition~\ref{unicque}. This proves
unique solvability of (\ref{pde}). For regularity of $v$ see
Lemma~\ref{urM}(a).
\qed

As we see from Theorem~\ref{thw2} and Proposition~\ref{unicque},
the notions of weak solution and $\omega$-solution coincide provided
$m>\max\{-p,-N\}$.

Using a slight modification it is also possible to
obtain existence and uniqueness results
for (\ref{pde}) with a weaker notion of strong solution: $v\in
C^2(B\setminus\{0\})$ instead of $v\in C^2(B)\cap C(\overline B)$,
that is, for solutions allowing singularity at $0\in B$.

\paragraph{Acknowledgments.} The third author expresses his deepest
gratitude to the International Centre for Theoretical Physics in
Trieste, Italy, for having had opportunity to participate
valuable courses and conferences
in Nonlinear Functional Analysis.

%%%%%%%%



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\end{thebibliography}  \medskip

\noindent{\sc Luka Korkut} (e-mail: luka.korkut@fer.hr) \\
{\sc  Mervan Pa\v si\'c} (e-mail: mervan.pasic@fer.hr) \\
{\sc Darko \v Zubrini\'c} (e-mail: darko.zubrinic@fer.hr) \\[2pt]
Department of mathematics \\
Faculty of Electrical Engineering and Computing \\
Unska 3, 10000 Zagreb, Croatia 
\end{document}