\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 11, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/11\hfil Riccati-type  inequality]
{Riccati-type inequality  and oscillation criteria for a half-linear 
 PDE with damping}

\author[Robert Ma\v r\'\i k \hfil EJDE-2004/11\hfilneg]
{Robert Ma\v r\'\i k}

\address{Mendel University, Department of Mathematics \\
 Zem\v ed\v elsk\'a 3 \\
 613 00 Brno, Czech Republic}
\email{marik@mendelu.cz\quad  phone: +420 545 134 030}

\date{}
\thanks{Submitted October 10, 2003. Published January 15, 2004.}
\subjclass[2000]{35J60, 35B05}
\keywords{$p$-Lapalacian, oscillatory solution, Riccati equation, \hfill\break\indent
 half-linear equation, damped equation, differential equation}

\begin{abstract}
  Under suitable conditions on the coefficients of a partial
  differential equation, we prove a Riccati-type inequality.
  As an application of this result, we find  oscillation
  criteria for second order damped half-linear partial differential
  equations. These criteria improve and complement
  earlier results on oscillation for partial  differential equations.
  The main feature in our results is that the oscillation criteria
  are not radially symmetric and do not depend only on the mean value
  of the coefficients. We consider unbounded domains and state a special
  oscillation criterion for conic domains.
\end{abstract}

\maketitle
 \numberwithin{equation}{section}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{corollary}[theorem]{Corollary}
 \newtheorem{definition}[theorem]{Definition}
 \newtheorem{remark}[theorem]{Remark}
 \newtheorem{example}[theorem]{Example}
\renewcommand{\theenumi}{\roman{enumi}}

\section{Introduction}

It is well known that the Riccati differential equation
\begin{equation}
  w'+w^2+c(x)=0\label{R}
\end{equation}
plays an important role in the study of the second order linear
differential equation
\begin{equation}
  u''+c(x)u=0.\label{SL}
\end{equation}
In fact, if \eqref{SL} has a positive solution $u$ on an interval
$I$, then the function $w=u'/u$ is a solution of \eqref{R},
defined on $I$. Conversely, if the Riccati inequality
\begin{equation*}%   \label{RN}
  w'+w^2+c(x)\leq 0
\end{equation*}
has a solution $w$, defined on $I$, then \eqref{SL} has a positive
solution on $I$. It is also well known that this property can be
extended also to other types of second order differential
equations and inequalities, which include the selfadjoint second
order differential equation
\begin{equation*}
  \bigl(r(x)u'\bigr)'+c(x)u=0\,,
\end{equation*}
the half-linear equation
\begin{equation}\label{PL}
  \bigl(r(x)|u'|^{p-2}u'\bigr)'+c(x)|u|^{p-2}u=0, \quad p>1,
\end{equation}
and the Schr\"odinger equation
\begin{equation*}%\label{LN}
 \sum_{i,j=1}^n\frac{\partial}{\partial x_i}\Bigl(a_{ij}(x)\frac{\partial
 u}{\partial x_j}\Bigr)+c(x)u=0\,.
\end{equation*}
see for example  \cite{B, N-S, Sch89,S66, S79}.

Another important fact of the substitution $w=u'/u$ for the
Riccati equation is that it is embedded in the Picone identity
which forms the link between the so-called Riccati technique and
variational technique in the oscillation theory of equation
\eqref{SL}. See  Section 3 for a short discussion concerning
the Picone identity.

In this paper we study the partial Riccati-type differential
inequality
\begin{equation*}
%   \label{eq:R1}
  \mathop{\rm div}\vec w+\|\vec w\|^q+c(x)\leq 0
\end{equation*}
and some generalizations of this inequality in the forms
\begin{equation}
  \label{eq:R2}
  \mathop{\rm div}(\alpha(x)\vec w)+K\alpha(x)\|\vec w\|^q+\alpha(x)c(x)\leq 0
\end{equation}
and
\begin{equation}
  \label{eq:R3}
  \mathop{\rm div}\vec w+K\|\vec w\|^q+c(x)+\langle \vec w, \vec b\rangle \leq0,
\end{equation}
where $K\in\mathbb{R}$, $q>1$. The assumptions on the functions $\alpha$, $b$
and $c$ are stated bellow. The operator $\mathop{\rm div} (\cdot)$
is the usual divergence operator, i.e. for $\vec w=(w_1,\dots, w_n)$
it holds
$\mathop{\rm div} \vec w=\sum_{i=1}^{n}\frac{\partial w_i}{\partial x_i}$,
 the norm
$\|\cdot\|$ is the usual Euclidean norm in $\mathbb{R}^n$ and
$\langle \cdot,\cdot\rangle$ is the usual scalar product in $\mathbb{R}^n$.

As an application of these results, we obtain new oscillation criteria for the
half-linear partial differential equation with damping.
The main difference between the criteria  obtained and similar results
in the literature lies in the fact, that our criteria are not
``radially symmetric''. See the discussion in Section 3, bellow.

This paper is organized as follows.  In the next section the
Riccati-type inequality is studied. The results of this section are
applied in the third section, which contains the results concerning
the oscillation for damped half-liner PDEs. The last
section is for examples and comments.

\section{Riccati inequality}

\subsection*{Notation} Let
\begin{gather*}
\Omega(a)=\{x\in\mathbb{R}^n:a\leq \|x\|\},\\
\Omega(a,b)=\{x\in\mathbb{R}^n:a\leq\|x\|\leq b\},\\
S(a)=\{x\in\mathbb{R}^n: \|x\|=a\}.
\end{gather*}
Let $p>1$ and $q>1$ be mutually conjugate numbers, i.e.
$1/p+1/q=1$. Let $\omega_n$ be the surface of the unit sphere in $\mathbb{R}^n$.
For $M \subseteq \mathbb{R}^n$, the symbols $\overline M$ and $M^0$ denote the
closure and the interior of $M$, respectively.


Integration over the domain $\Omega(a,b)$ is performed introducing
hyperspherical coordinates $(r,\theta)$, i.e.
\begin{equation*}
  \int_{\Omega(a,b)}f(x)\,\mathrm{d}x=\int_a^b\int_{S(r)}f(x(r,\theta))\,\mathrm{d}S\,\mathrm{d}r,
\end{equation*}
where $\,\mathrm{d}S$ is the element of the surface of the sphere $S(r)$.

We will study the Riccati inequality on two
types of unbounded domains in $\mathbb{R}^n$: The exterior
of a ball, centered in the origin, and a general unbounded domain
$\Omega$. In the latter case we use the assumption:
\begin{itemize}
\item[(A1)] The set $\Omega$ is an unbounded domain in $\mathbb{R}^n$, simply
  connected with a piecewise smooth boundary $\partial \Omega$ and
  $\mathop{\rm meas} (\Omega\cap S(t))>0$ for $t>1$.
\end{itemize}

\begin{theorem}\label{L2}
  Let $\Omega$ satisfy  \textup{(A1)} and $c\in
  C(\Omega,\mathbb{R})$. Suppose $\alpha$ satisfies
  \begin{gather}
    \alpha\in C^1(\Omega\cap\Omega(a_0),\mathbb{R}^+)\cap
    C_0(\overline\Omega, \mathbb{R}), \nonumber %
% \label{eq:P100a} 
\\
    \label{eq:P1a}
    \int_{a_0}^\infty\Bigl(\int _{\Omega\cap S(t)}\alpha
    (x)\,\mathrm{d}S\Bigr)^{1-q}\,\mathrm{d}t=\infty\,.
  \end{gather}
  Also suppose that there exist $a\geq a_0$, a real constant $K>0$
  and a real-valued differentiable vector function $\vec w(x)$ which
  is bounded (in the sense of the continuous extension, if necessary)
  on every compact subset of $\overline{\Omega\cap \Omega(a)}$ and
  satisfies the differential inequality \eqref{eq:R2}
  on $\Omega\cap\Omega(a)$. Then
  \begin{equation}
    \label{eq:P2a}
    \liminf_{t\to\infty}\int_{\Omega\cap\Omega(a_0,t)}\alpha(x)c(x)\,\mathrm{d}x<\infty.
  \end{equation}
\end{theorem}

\begin{proof}
  For simplicity let us denote $\tilde\Omega(a)=\Omega(a)\cap\Omega$,
  $\tilde S(a)=S(a)\cap\Omega$, $\tilde\Omega(a,b)=\Omega(a,b)\cap
  \Omega$.  Suppose, by contradiction, that %\eqref{eq:P0a},
  \eqref{eq:P1a} and \eqref{eq:R2}
  are fulfilled and
  \begin{equation}
    \lim_{t\to\infty}\int_{\tilde\Omega(a_0,t)}\alpha(x)c(x)\,\mathrm{d}x=
    \infty. \label{eq:*}
  \end{equation}
  Integrating \eqref{eq:R2} over
  the domain $\tilde\Omega(a,t)$ and applying the
  Gauss-Ostrogradski divergence theorem gives
  \begin{multline}
    \label{L2:0}
    \int_{\tilde S(t)}\alpha(x)\langle \vec w(x),\vec\nu(x)\rangle\,\mathrm{d}S
    -\int_{\tilde S(a)}\alpha(x)\langle \vec w(x),\vec \nu(x)\rangle\,\mathrm{d}S \\
    +\int_{\tilde \Omega(a,t)}\alpha(x)c(x)\,\mathrm{d}x+K\int_{\tilde
      \Omega(a,t)} \alpha(x)\|\vec w(x)\|^q\,\mathrm{d}x\leq 0,
  \end{multline}
  where $\vec\nu(x)$ is the outside normal unit vector to the sphere
  $S(\|x\|)$ in the point $x$ (note that the product $\alpha (x)\vec
  w(x)$ vanishes on the boundary $\partial\Omega$ since
  $\alpha\in C_0(\overline \Omega,\mathbb{R})$ and $\vec w$ is bounded near the
  boundary).
  In view of \eqref{eq:*} there exists $t_0\geq a$ such that
  \begin{equation}
    \label{L2:1}
    \int_{\tilde \Omega(a,t)}\alpha(x)c(x)\,\mathrm{d}x-\int_{\tilde
    S(a)}\alpha(x)\langle \vec w(x),\vec\nu(x)\rangle \,\mathrm{d}S \geq 0
  \end{equation}
  for every $t\geq t_0$. Further Schwarz and H\"older inequality give
  \begin{align}\label{L2:2}
  -\int_{\tilde S(t)}\alpha(x)\langle \vec w(x),\vec\nu(x)\rangle \,\mathrm{d}S
  &\leq   \int_{\tilde  S(t)}\alpha(x)\|w(x)\|\,\mathrm{d}S \nonumber\\
   & \leq \Bigl(\int_{\tilde S(t)}\alpha(x)\|w(x)\|^q\,\mathrm{d}S\Bigr)^{1/q}
   \Bigl(\int_{\tilde S(t)}\alpha(x)\,\mathrm{d}S\Bigr)^{1/p}.
  \end{align}
  Combination of inequalities  \eqref{L2:0}, \eqref{L2:1}, and \eqref{L2:2}
gives
  \begin{equation*}
    K\int_{\tilde \Omega(a,t)}\alpha(x)\|\vec w (x)\|^q\,\mathrm{d}x\leq
    \Bigl(\int_{\tilde S(t)}\alpha(x)\|w(x)\|^q\,\mathrm{d}S
    \Bigr)^{1/q}\Bigl( \int_{\tilde S(t)}\alpha
    (x)\,\mathrm{d}S\Bigr)^{1/p}
  \end{equation*}
  for every $t\geq t_0$. Denote
  \begin{equation*}
    g(t)=\int_{\tilde \Omega(a,t)}\alpha(x)\|w(x)\|^q\,\mathrm{d}x.
  \end{equation*}
  Then the last inequality can be written in the form
  \begin{equation*}
    Kg(t)\leq\Bigl(g'(t)\Bigr)^{1/q}\Bigl(\int_{\tilde S(t)}\alpha(x)\,\mathrm{d}S
    \Bigr)^{1/p}.
  \end{equation*}
  From here we conclude for every $t\geq t_0$
  \begin{equation*}
    K^qg^q(t)\leq g'(t)\Bigl(\int_{\tilde
    S(t)}\alpha(x)\,\mathrm{d}S\Bigr)^{q/p}
  \end{equation*}
  and equivalently
  \begin{equation*}
    K^q\Bigl(\int_{\tilde S(t)}\alpha(x)\,\mathrm{d}S\Bigr)^{1-q}\leq
    \frac{g'(t)}{g^q(t)}.
  \end{equation*}
  This inequality shows that the integral on the left-hand side of
  \eqref{eq:P1a} has an integrable majorant on $[t_0,\infty)$ and
  hence it is convergent as well, a contradiction to \eqref{eq:P1a}.
\end{proof}

A commonly considered case is $\Omega=\mathbb{R}^n$ or $\Omega=\Omega(a_0)$. In
this case the preceding theorem gives:

\begin{theorem}\label{L3}
  Let $\alpha\in C^1(\Omega(a_0),\mathbb{R}^+)$, $c\in C(\Omega(a_0),
  \mathbb{R})$. Suppose that
  \begin{equation}
    \label{eq:P1}
    \int_{a_0}^\infty\Bigl(\int _{S(t)}\alpha
    (x)\,\mathrm{d}S\Bigr)^{1-q}\,\mathrm{d}t=\infty .
  \end{equation}
  Further suppose, that there exists $a\geq a_0$, real constant $K>0$
  and real--valued differentiable vector function $\vec w(x)$ defined
  on $\Omega(a)$ which satisfies the differential inequality
  \eqref{eq:R2} on $\Omega(a)$.
  Then
  \begin{equation}
    \label{eq:P2}
    \liminf_{t\to\infty}\int_{\Omega(a_0,t)}\alpha(x)c(x)\,\mathrm{d}x<\infty.
  \end{equation}
\end{theorem}

  The proof of this theorem is a modification and simplification of the proof of
  Theorem \ref{L2}.



In the following theorem we will use the integral averaging technique
which is due to Philos \cite{Ph}, where the linear ordinary
differential equation is considered. This technique has been later
extended in several directions (see \cite{Li,M-AMC, Wang} and the
references therein).  The main idea of this technique is in the
presence of the two-parametric weighting function $H(t,x)$ defined on
the closed domain
\begin{equation*}
  D=\{(t,x)\in\mathbb{R}\times\mathbb{R}^n: a_0\leq \|x\|\leq t\}
\end{equation*}
Further denote $D_0=\{(t,x)\in\mathbb{R}\times\mathbb{R}^n: a_0<\|x\|< t\}$ and
suppose that the function $H(t,x)$ satisfies the hypothesis
\begin{itemize}
\item[(A2)] $H(t,x)\in C(D,\mathbb{R}_0^+)\cap C^1(D_0, \mathbb{R}_0^+)$.
\end{itemize}
Some additional assumptions on the function $H$ are stated bellow.
First let us remind the well-known Young inequality.

\begin{lemma}[Young inequality]
  For $\vec a,\vec b\in\mathbb{R}^n$
  \begin{equation}\label{eq:YoungIneq}
    \frac{\|\vec a\|^p}p\pm\langle \vec a, \vec b\rangle
    +\frac{\|\vec b\|^q}q\geq 0\,.
  \end{equation}
\end{lemma}

\begin{theorem}\label{L4}
  Let $\Omega$ be an unbounded domain in $\mathbb{R}^n$ which satisfies \textup{(A1)},
  $c\in C(\Omega,\mathbb{R})$ and $\vec b\in  C(\Omega, \mathbb{R}^n)$. Suppose the  function
  $H(t,x)$ satisfies \textup{(A2)} and the following conditions:
  \begin{enumerate}
  \item $H(t,x)\equiv 0$ for $x\not\in \overline\Omega$.
  \item If $x\in\partial\Omega$, then $H(t,x)=0$ and $\|\nabla H(t,x)\|=0$
  for every $t\geq x$.
  \item If $x\in \Omega^0$, then $H(t,x)=0$ if and only if $\|x\|=t$.
  \item The vector function $\vec h(x)$ defined on $D_0$ with the
    relation
    \begin{equation}
      \label{eq:h}
      \vec h(t,x)=-\nabla H(t,x)+\vec b(x)H(t,x)
    \end{equation}
    satisfies
    \begin{equation}
      \label{eq:konv-h}
      \int_{\Omega(a_0,t)\cap\Omega}H^{1-p}(t,x)\|\vec h(t,x)\|^p\,\mathrm{d}x<\infty.
    \end{equation}
  \item There exists a continuous function $k(r)\in C([a_0,\infty),
    \mathbb{R}^+)$ such that the function $\Phi(r):= k(r)
    \int_{S(r)\cap\Omega}H(t,x)\,\mathrm{d}x$ is positive and nonincreasing on
    $[a_0,t)$ with respect to the variable $r$ for every $t$, $t>r$.
  \end{enumerate}
  Also suppose that there exist real numbers $a\geq a_0$, $K>0$
  and differentiable vector function $\vec w(x)$ defined on $\Omega$
  which is bounded on every compact subset of
  $\overline{\Omega\cap\Omega(a)}$ and satisfies the Riccati inequality
  \eqref{eq:R3} on $\Omega\cap\Omega(a)$.
  Then
  \begin{equation}
    \label{eq:limsup}
    \limsup_{t\to\infty}\Bigl(\int_{S(a_0)}H(t,x)\,\mathrm{d}S\Bigr)^{-1}
    \int\limits_{\Omega(a_0,t)\cap\Omega}\Bigl[H(t,x)c(x)
    -\frac{\|\vec h(t,x)\|^p}{(Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x<\infty
  \end{equation}
\end{theorem}

\begin{remark} \rm
  Let us mention that nabla operator $\nabla H(t,x)$ relates
  only to the variables of $x$, i.e.
  $\nabla  H(t,x)=(\frac{\partial}{\partial x_1}, \dots,
  \frac{\partial}{\partial x_n})H(t,x)$, and does not relate to the
  variable $t$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{L4}]
For simplicity let us  introduce the notation $\tilde \Omega(a)$, $\tilde S(a)$
and $\tilde  \Omega(a,b)$ as in the proof of Lemma \ref{L2}.  Suppose that the
  assumptions of theorem are fulfilled. Multiplication
  of \eqref{eq:R3} by the function $H(t,x)$ gives
  \[
    H(t,x)\mathop{\rm div} \vec w(x)+H(t,x)c(x)+KH(t,x)
    \|\vec w(x)\|^q+H(t,x)\langle \vec w(x) ,\vec b(x)\rangle \leq 0
  \]
  and equivalently
  \begin{multline*}
\mathop{\rm div} (H(t,x)\vec w(x))+H(t,x)c(x)\\
+KH(t,x) \|\vec w(x)\|^q+\langle \vec w(x),H(t,x)\vec b(x)-\nabla H(t,x)\rangle\leq 0
  \end{multline*}
  for $x\in\tilde\Omega(a)$ and $t\geq \|x\|$.  This and Young
  inequality \eqref{eq:YoungIneq} implies
  \begin{equation*}
    \mathop{\rm div} (H(t,x)\vec w(x))+H(t,x)c(x)
    -\frac{\|H(t,x)\vec b(x)-\nabla  H(t,x)\|^p}{ (Kq)^{p-1}pH^{p-1}(t,x)}\leq 0.
  \end{equation*}
  Integrating this inequality over the domain $\tilde\Omega(a,t)$
  and the Gauss-Ostrogradski divergence theorem give
\[
    -\int_{\tilde S(a)}H(t,x)\langle \vec w(x), \vec \nu(x)\rangle \,\mathrm{d}S
     +\int_{\tilde \Omega(a,t)}\Bigl[H(t,x)c(x)
- \frac{\|\vec h(t,x)\|^p}{(Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x\leq 0
\]
  and hence
  \begin{equation*}
    \int_{\tilde \Omega(a,t)}\Bigl[H(t,x)c(x)
  - \frac{\|\vec h(t,x)\|^p}{ (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x
  \leq \int_{\tilde   S(a)}H(t,x)\|w(x)\|\,\mathrm{d}S
  \end{equation*}
  holds for $t>a$. This bound we will use to estimate the integral from
  the condition \eqref{eq:limsup}
  \begin{align*}
    &\int_{\tilde \Omega(a_0,t)}\Bigl[H(t,x)c(x)
    - \frac{\|\vec h(t,x)\|^p}{ (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x\\
    &=\int_{\tilde \Omega(a_0,a)}\Bigl[H(t,x)c(x)
    - \frac{\|\vec h(t,x)\|^p}{ (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x\\
  &\quad + \int_{\tilde \Omega(a,t)}\Bigl[H(t,x)c(x)
   - \frac{\|\vec h(t,x)\|^p}{(Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x\\
  &\leq  \int_{\tilde\Omega(a_0,a)}H(t,x)c(x)\,\mathrm{d}x+\int_{\tilde
        S(a)}H(t,x)\|w(x)\|\,\mathrm{d}S.
  \end{align*}
  Denote the maximal functions $c^*(r)=\max\{|c(x)|:x\in S(r)\}$ and
  $w^*(r)=\max\{\|w(x)\|:x\in S(r)\}$. Then it holds
  \begin{align*}
  &\int_{\tilde\Omega(a_0,t)}\Bigl[H(t,x)c(x)
  - \frac{\|\vec h(t,x)\|^p}{(Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x     \\
  &\leq \int_{a_0}^{a}\Bigl[ k(r)\int_{\tilde S(r)}H(t,x)\,\mathrm{d}S\Bigr]
  \frac{c^*(r)}{k(r)}\,\mathrm{d}r
  +k(a)\frac{w^*(a)}{ k(a)}\int_{\tilde S(a)}H(t,x)\,\mathrm{d}S \\
 & \leq k(a_0)\int_{\tilde S(a_0)}H(t,x)\,\mathrm{d}S\Big[\int_{a_0}^a
 \frac{c^*(r)}{k(r)}\,\mathrm{d}r +\frac{w^*(a)}{k(a)}\Big]
  \end{align*}
  for every $t\geq a_0$. From here we conclude that the expression
  \begin{equation*}
    \Bigl(\int_{\tilde S(a_0)}H(t,x)\,\mathrm{d}S\Bigr)^{-1}
    \int_{\tilde\Omega(a_0,t)}\Bigl[H(t,x)c(x)
    -\frac{\|\vec h(t,x)\|^p}{(Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x
  \end{equation*}
  is bounded for all $t\geq a_0$. Hence \eqref{eq:limsup} follows. The
  proof is complete.
\end{proof}

As in Theorem \ref{L3}, we state the result of Theorem \ref{L4} also
for $\Omega=\mathbb{R}^n$.

\begin{theorem}\label{L5}
  Let $c\in C(\Omega(a_0))$, $\vec b\in C(\Omega(a_0), \mathbb{R}^n)$. Suppose that
  the function $H(t,x)$ satisfies hypothesis \textup{(A2)} and the
  following conditions:
  \begin{enumerate}
  \item $H(t,x)=0$ if and only if $\|x\|=t$
  \item The vector function $\vec h(x)$ defined on $D_0$ with the
    relation \eqref{eq:h}   satisfies
    \begin{equation*}
%       \label{eq:konv-h-2}
      \int_{\Omega(a_0,t)}H^{1-p}(t,x)\|\vec h(t,x)\|^p\,\mathrm{d}x<\infty
    \end{equation*}
  \item There exists a continuous function $k(r)\in C([a_0,\infty),
    \mathbb{R}^+)$ such that the function $\Phi(r):=k(r)\int_{S(r)}H(t,x)\,\mathrm{d}x$
    is positive and nonincreasing on $[a_0,t)$ with respect to the
    variable $r$ for every $t$, $t>r$.
  \end{enumerate}
Further suppose that there exist real numbers $a\geq a_0$, $K>0$
and differentiable vector function $\vec w(x)$ defined on $\Omega(a)$
which satisfies the Riccati inequality \eqref{eq:R3} on $\Omega(a)$.
Then
\begin{equation*}
%   \label{eq:limsup-2}
  \limsup_{t\to\infty}\Bigl(\int_{S(a_0)}H(t,x)\,\mathrm{d}S\Bigr)^{-1}
  \int_{\Omega(a_0,t)}\Bigl[H(t,x)c(x)
  -\frac{\|\vec h(t,x)\|^p}{(Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x<\infty
\end{equation*}
\end{theorem}

  The proof of this theorem is a simplification of the proof of Theorem  \ref{L4}.

\section{Oscillation for half-linear equation}

In this section we will employ the results concerning the Riccati
inequality to derive oscillation criteria for the second order partial
differential equation
\begin{equation}  \label{eq:E}
  \mathop{\rm div}(\|\nabla u\|^{p-2}\nabla u)
  +\langle \vec b(x),  \|\nabla u\|^{p-2}\nabla u\rangle +c(x)|u|^{p-2}u=0,
\end{equation}
where $p>1$.  The second order differential operator $\mathop{\rm
div}(\|\nabla u\|^{p-2}\nabla u)$ is called the $p$-Laplacian
and this operator is important in various technical applications
and physical problems -- see \cite{Diaz}. The functions $c$ and
$\vec b$ are assumed to be H\"older continuous functions on the
domain $\Omega(1)$. The solution of \eqref{eq:E} is every function
defined on $\Omega(1)$ which satisfies \eqref{eq:E} everywhere on
$\Omega(1)$.

The special cases of equation \eqref{eq:E} are the linear equation
\begin{equation}
  \label{eq:L}
  \Delta u+\langle\vec b,\nabla u\rangle +c(x)u=0
\end{equation}
which can be obtained for $p=2$, the Schr\"odinger equation
\begin{equation}
  \label{eq:Schr}
  \Delta u+c(x)u=0
\end{equation}
obtained for $p=2$ and $\vec b\equiv 0$
and the undamped half-linear equation
\begin{equation}
  \label{eq:ENetl}
  \mathop{\rm div}(\|\nabla u\|^{p-2}\nabla u)+c(x)|u|^{p-2}u=0
\end{equation}
for $\vec b\equiv 0$.

Equation \eqref{eq:E} is called the \textit{half-linear equation},
since the operator on the left-hand side is homogeneous and hence
a constant multiple of every solution of \eqref{eq:E} is a
solution of \eqref{eq:E} as well. If $p=2$, then equation
\eqref{eq:E} is linear elliptic equation \eqref{eq:L}, however in
the general case $p\neq 2$ is the linearity of the space of
solutions lost and only homogenity remains.

Concerning the linear equation two types of oscillation are studied
-- \textit{nodal} oscillation and \textit{strong} oscillation. The
equivalence between these two types of oscillation has been proved in
\cite{M-P78} for locally H\"older continuous function $c$, which is an
usual assumption concerning the smoothness of $c$, see also \cite{DM}
for short discussion concerning the general situation $p\neq 2$.  In
the connection to equation \eqref{eq:E} we will use the following concept
of oscillation.

\begin{definition} \rm
  The function $u$ defined on $\Omega(1)$ is said to be
  \textit{oscillatory}, if the set of the zeros of the function $u$ is
  unbounded with respect to the norm.  Equation \eqref{eq:E} is said
  to be \textit{oscillatory} if every its solution defined on
  $\Omega(1)$ is oscillatory.
\end{definition}

\begin{definition} \rm
  Let $\Omega$ be an unbounded domain in $\mathbb{R}^n$.  The function $u$
  defined on $\Omega(1)$ is said to be \textit{oscillatory in the
    domain $\Omega$}, if the set of the zeros of the function $u$,
  which lies in the closure $\overline \Omega$ is unbounded with
  respect to the norm. Equation \eqref{eq:E} is said to be
  \textit{oscillatory in the domain $\Omega$} if every its solution
  defined on $\Omega(1)$ is oscillatory in $\Omega$.  The equation is
  said to be \textit{nonoscillatory} (\textit{nonoscillatory in
    $\Omega$}) if it is not oscillatory (oscillatory in $\Omega)$.
\end{definition}

Due to the homogenity of the set of solutions, it follows from the
definition that the equation which possesses a solution on $\Omega(1)$
is nonoscillatory, if it has a solution $u$ which is positive on
$\Omega(T)$ for some $T>1$ and oscillatory otherwise. Further the
equation is nonoscillatory in $\Omega$ if it has a solution $u$ such
that $u$ is positive on $\overline\Omega\cap\Omega(T)$ for some $T>1$
and oscillatory otherwise.

Jaro\v s et. al. studied in \cite{JKY} the partial differential equation
\begin{equation}
  \label{eq:E-JKY}
  \mathop{\rm div}\bigl(a(x)\|\nabla u\|^{p-2}\nabla u\bigr)+c(x)|u|^{p-2}u=0,
\end{equation}
where $a(x)$ is a positive smooth function and obtained the
Sturmian-types comparison theorems and oscillation criteria for
\eqref{eq:E-JKY}. The same results have been proved independently by
Do\v sl\'y and Ma\v r\'\i k in \cite{DM} for the case $a(x)\equiv 1$.

\begin{theorem}[\cite{DM,JKY}]\label{th:JKY}
  Equation \eqref{eq:E-JKY} is oscillatory, if the ordinary
  differential equation
  \begin{equation*}
    (r^{n-1}\overline a(r)|y'|^{p-2}y')'+r^{n-1}\overline
    c(r)|y|^{p-2}y=0, \quad {}'=\frac{\hbox{\rm d}}{\,\mathrm{d}r}
  \end{equation*}
  is oscillatory, where $\overline a(r)$ and $\overline c(r)$ denote
  the mean value of the function $a$ and $c$ over the sphere $S(r)$,
  respectively, i.e.
  \begin{equation*}
    \overline a(r)=\frac {1}{ \omega_nr^{n-1}}\int_{S(r)}a(x)\,\mathrm{d}S, \quad
    \overline c(r)=\frac{1}{ \omega_nr^{n-1}}\int_{S(r)}c(x)\,\mathrm{d}S.
  \end{equation*}
\end{theorem}

The main tool in the proof of this theorem is a Picone identity
for equation \eqref{eq:E-JKY}.  Another application (not only to
the oscillation or comparison theory) of the Picone identity to
the equation with $p$-Lapalacian can be found in \cite{AH}.

Concerning the Riccati-equation methods in the oscillation theory of
PDE, Noussair and Swanson used in \cite{N-S} the  transformation
\begin{equation*}
  \vec w(x)=-\frac{\alpha(\|x\|)}{\phi(u)}(A\nabla u)(x)
\end{equation*}
to detect nonexistence of eventually positive solution of the
semilinear inequality
\begin{equation*}
  \sum _{i,j=1}^n \frac{\partial}{\partial
  x_i}\Bigl(a_{ij}(x)\frac{\partial u}{\partial
  x_j}\Bigr)+p(x)\phi(u)\leq 0,
\end{equation*}
which seems to be one of the first papers concerning the
transformation of PDE into the Riccati type equation.

In the paper of Schminke \cite{Sch89} is the Riccati technique used in
the proof of nonexistence of positive and eventually positive solution
of Schr\"odinger equation \eqref{eq:Schr}. The results are expressed
in the spectral terms, concerning the lower spectrum of Schr\"odinger
operator.


Recently Kandelaki et. al. \cite{B} via the Riccati technique improved
the Nehari and Hille criteria for oscillation and nonoscillation of
linear second order equation \eqref{SL} and extended these criteria to
the half-linear equation \eqref{PL}.  The further extension of the
oscillatory results from \cite{B} to the case of equation
\eqref{eq:ENetl} can be found in \cite{M2}. One of the typical result
concerning the oscillation of equation \eqref{eq:ENetl} is the
following.

\begin{theorem}[Hartman--Wintner type criterion, \cite{M1}]\label{th:HW}
  Denote
  \begin{equation*}
    C(t)=\frac{p-1}{ t^{p-1}}\int_1^t s^{p-2}\int_{\Omega(1,s)}
    \|x\|^{1-n}c(x)\,\mathrm{d}x\,\mathrm{d}s.
  \end{equation*}
  If
  \begin{equation*}
    -\infty<\liminf_{t\to\infty}C(t) < \limsup_{t\to\infty}C(t)
    \leq \infty
    \quad\text{ or if  }\quad
    \lim_{t\to\infty}C(t)=\infty,
  \end{equation*}
  then equation \eqref{eq:ENetl} is oscillatory.
\end{theorem}

A quick look at this condition and also at Theorem \ref{th:JKY} reveals
that the potential function $c(x)$ is in these criteria contained only
within the integral over the balls, centered in the origin. As a
consequence of this fact it follows that though the criteria are sharp
in the cases when the function $c(x)$ is radially symmetric, these
criteria cannot detect the contingent oscillation of the equation in
the cases when the mean value of the function $c(x)$ over the balls
centered in the origin is small. In order to remove this disadvantage
we will apply the theorems from the preceding section to the Riccati
equation obtained by the transformation of equation \eqref{eq:E}. As
a result we obtain the oscillation criteria which are applicable also
in such extreme cases when $\int_{S(r)} c(x)\,\mathrm{d}S=0$. The criteria can
detect also the oscillation over the more general exterior domains, than
the exterior of some ball. An application to the oscillation over the
conic domain is given in Section 4.

Remark that there are only few results in the literature concerning
the oscillation on another types of unbounded domain, than an exterior
of a ball. Let us mention the paper of Atakarryev and Toraev
\cite{A-T86}, where Kneser--type oscillation criteria for various
types of unbounded domains were derived for the linear equation
\begin{equation*}
  \sum _{i,j=1}^n a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial
  x_j}+p(x)u=0.
\end{equation*}
In the paper \cite{JKY2} of Jaro\v s et. al. the forced superlinear
equation
\begin{equation*}
  \sum_{i,j=1}^n\frac{\partial}{\partial
  x_i}\Bigl(a_{ij}(x)\frac{\partial u}{\partial
  x_j}\Bigr)+c(x)|u|^{\beta-1}u=f(x),\quad \beta>1
\end{equation*}
is studied via the Picone identity and the results concerning
oscillation on the domains with piecewise smooth boundary are established.

Our main tool will be the following Lemma \ref{Le:3.1} which presents
the relationship between positive solution of \eqref{eq:E} and a
solution of the Riccati--type equation.

\begin{lemma}\label{Le:3.1}
  Let $u$ be solution of \eqref{eq:E} positive on the domain $\Omega$.
  The vector function $\vec w(x)$ defined by
  \begin{equation}
    \label{eq:RicSub}
    \vec w(x)=\frac{\|\nabla u(x)\|^{p-2}\nabla u(x)}{|u(x)|^{p-2}u(x)}
  \end{equation}
  is well defined on $\Omega$ and satisfies the Riccati equation
  \begin{equation}
    \label{eq:RicRce}
    \mathop{\rm div}\vec w+c(x)+(p-1)\|\vec w\|^q
  +\langle \vec w ,\vec b(x)\rangle =0
  \end{equation}
  for every $x\in\Omega$.
\end{lemma}

\begin{proof} From \eqref{eq:RicSub} it follows (the dependence on the
  variable $x$ is suppressed in the notation)
  \begin{equation*}
    \mathop{\rm div} \vec w=\frac{\mathop{\rm div}(\|\nabla u\|^{p-2}\nabla
    u)}{|u|^{p-2}u}- (p-1)\frac{\|\nabla u\|^{p}}{|u|^p}
  \end{equation*}
  on the domain $\Omega$.  Since $u$ is a positive solution of
  \eqref{eq:E} on $\Omega$ it follows
  \begin{align*}
    \mathop{\rm div} \vec w
  &= -c- \langle \vec b,\frac{\|\nabla u\|^{p-2}\nabla u}{|u|^{p-2}u}\rangle
  -(p-1) \frac{\|\nabla u\|^{p}}{|u|^p}\\
  &=- c-(p-1)\frac{\|\nabla u\|^{p}}{|u|^p}
  -\langle \vec  b, \frac{\|\nabla u\|^{p-2}\nabla u}{|u|^{p-2}u}\rangle.
  \end{align*}
  Application of \eqref{eq:RicSub} gives
$ \mathop{\rm div} \vec w=- c-(p-1)\|\vec w\|^{q}
-\langle \vec b, \vec w\rangle$
  on $\Omega$. Hence \eqref{eq:RicRce} follows.
\end{proof}

The first theorem concerns the case in which left-hand sides of
\eqref{eq:RicRce} and \eqref{eq:R2} differ only in a multiple
by the function $\alpha$.

\begin{theorem}\label{Th1}
  Suppose that there exists function $\alpha\in C^1(\Omega(a_0),
  \mathbb{R}^+)$ which satisfies
  \begin{enumerate}
  \item  for $x\in\Omega(a_0)$
    \begin{equation}
      \label{eq:P3}
      \nabla \alpha(x)=\vec b(x)\alpha(x)
    \end{equation}
  \item the condition \eqref{eq:P1} holds and
  \item
    \begin{equation}
      \label{eq:P2n}
      \lim_{t\to\infty}\int_{\Omega(a_0,t)}\alpha(x)c(x)\,\mathrm{d}x=\infty.
    \end{equation}
  \end{enumerate}
  Then equation \eqref{eq:E} is oscillatory in $\Omega(a_0)$.
\end{theorem}

\begin{proof}
  Suppose, by contradiction, that \eqref{eq:P1}, \eqref{eq:P3} and
  \eqref{eq:P2n} hold and \eqref{eq:E} is not oscillatory in
  $\Omega(a_0)$. Then there exists a real number $a\geq a_0$ such that
  equation \eqref{eq:E} possesses a solution $u$ positive on
  $\overline\Omega(a)$. The function $\vec w(x)$ defined on
  $\Omega(a)$ by \eqref{eq:RicSub} is well-defined, satisfies
  \eqref{eq:RicRce} on $\Omega(a)$ and is bounded on every compact
  subset of $\overline\Omega(a)$. In view of the condition
  \eqref{eq:P3} equation \eqref{eq:RicRce} can be written in the form
  \begin{equation*}
    \alpha \mathop{\rm div} \vec w+\alpha c+(p-1)\alpha \|\vec w\|^q
    +\langle \vec w, \nabla \alpha\rangle =0
  \end{equation*}
  which implies \eqref{eq:R2} with $K=p-1$.  Theorem \ref{L3} shows
  that \eqref{eq:P2} holds, a contradiction to \eqref{eq:P2n}.
\end{proof}

The following theorem concerns the linear case $p=2$.

\begin{theorem}\label{th:lin}
  Let $\alpha\in C(\Omega(a_0),\mathbb{R}^+)$ Denote
  \begin{equation*}
%     \label{eq:C(x)}
    C_1(x)=c(x)-\frac 1{4\alpha^2(x)}\|\alpha(x)\vec b(x)-\nabla\alpha(x)\|^2
     -\frac 1{2\alpha(x)}\mathop{\rm div}\Bigl(\alpha(x)\vec
    b(x)-\nabla\alpha(x)\Bigr).
  \end{equation*}
  Suppose that
  \begin{align}
%     \label{eq:P6}
    \int_{a_0}^\infty\Bigl(\int_{S(t)}\alpha(x)\,\mathrm{d}S\Bigr)^{-1}\,\mathrm{d}t=\infty\,,
    \nonumber \\
    \label{eq:P7}
    \lim_{t\to\infty}\int_{\Omega(a_0,t)}\alpha(x)C_1(x)\,\mathrm{d}x=\infty.
  \end{align}
  Then equation \eqref{eq:L} is oscillatory in $\Omega(a_0)$.
\end{theorem}

\begin{proof}
  Suppose, by contradiction, that \eqref{eq:L} is nonoscillatory. As
  in the proof of Theorem \ref{Th1}, there exists $a\geq a_0$ such
  that \eqref{eq:RicRce} with $p=2$ has a solution $\vec w(x)$ defined
  on $\Omega (a)$.  Denote $\vec W(x)=\vec w(x)+\frac12\Bigl(\vec
  b-\frac{\nabla \alpha}{\alpha}\Bigr)$. Direct computation shows that
  the function $\vec W$ satisfies the differential equation
  \begin{equation*}
    \mathop{\rm div}\vec W+C_1(x)+\|\vec w\|^2+\bigl\langle\frac{\nabla
    \alpha}\alpha, \vec W\bigr\rangle=0
  \end{equation*}
  on $\Omega(a)$.
  From here we conclude that the function $\vec W$ satisfies
  \begin{equation*}
    \mathop{\rm div}(\alpha \vec W)+C_1\alpha+\alpha\|\vec W\|^2=0
  \end{equation*}
  on $\Omega (a)$. However by Theorem \ref{L3} inequality
  \eqref{eq:P2} with $C_1$ instead of $c$ holds, a contradiction to
  \eqref{eq:P7}.
\end{proof}

The next theorem concerns the general case $p>1$.
In this case we allow also another types of unbounded domains,
than $\Omega(a_0)$.

\begin{theorem}\label{T3}
  Let $\Omega$ be an unbounded domain which satisfies
  \textup{(A1)}. Suppose that $k\in(1,\infty)$ is a real number and
  $\alpha\in C^1(\Omega(a_0), \mathbb{R}_0^+)$ is a function defined on
  $\Omega(a_0)$ such that
  \begin{enumerate}
  \item $\alpha(x)=0$ if and only if $x\not\in\Omega\cap\Omega(a_0)$
  \item \eqref{eq:P1a} holds.
\end{enumerate}
  For $x\in\Omega\cap\Omega(a_0)$ denote
  \begin{equation*}
%     \label{eq:C2}
    C_2(x)=c(x)-\frac k{(p\alpha(x))^p}\|\alpha(x)\vec b(x)-\nabla\alpha(x)\|^p.
  \end{equation*}
  If
  \begin{equation}
    \label{eq:P8}
    \lim_{t\to\infty} \int_{\Omega\cap\Omega(a_0,t)}\alpha(x)C_2(x)\,\mathrm{d}x
    =\infty
  \end{equation}
  holds, then \eqref{eq:E} is oscillatory in $\Omega$.
\end{theorem}

\begin{remark} \rm
  Under \eqref{eq:P8} we understand that the integral
  \begin{equation*}
    f(t)=\int_{\Omega\cap S(t)}\alpha(x)C_2(x)\,\mathrm{d}S
  \end{equation*}
  which may have singularity near the boundary $\partial\Omega$ is
  convergent for large $t$'s and the function $f$ satisfy $\int^\infty
  f(t)\,\mathrm{d}t=\infty$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{T3}]
  Suppose, by contradiction, that \eqref{eq:E} is not oscillatory.
  Then there exists a number $a\geq a_0$ and a function $u$ defined on
  $\Omega(a)$ which is positive on $\overline{\Omega\cap\Omega(a)}$
  and satisfies \eqref{eq:E} on $\Omega\cap\Omega(a)$. The vector
  function $\vec w(x)$ defined by \eqref{eq:RicSub} satisfies
  \eqref{eq:RicRce} on $\Omega\cap\Omega(a)$ and is bounded on every
  compact subset of $\overline{\Omega\cap\Omega(a)}$. Denote
  $l=k^{\frac{1}{ p-1}}$ and let $l^*$ be a conjugate number to the number
  $l$, i.e.  $\frac{1}{l}+\frac{1}{ l^*}=1$ holds.  Clearly $l>1$ and
  $l^*>1$. The Riccati equation \eqref{eq:RicRce} can be written in
  the form
  \begin{equation*}
    \mathop{\rm div}\vec w+c(x)+\frac{p-1}{ l}\|\vec w\|^q
    +\langle \vec w ,\vec b(x)-\frac{\nabla \alpha}\alpha\rangle+
    \frac{p-1}{ l^*}\|\vec w\|^q+\langle \vec w,\frac{\nabla\alpha}{\alpha}\rangle=0
  \end{equation*}
  for $x\in\Omega\cap\Omega(a)$.
  From inequality \eqref{eq:YoungIneq} it follows
  \begin{align*}
    \frac{p-1}{l}\|\vec w\|^q+\langle \vec w, \vec b-\frac{\nabla\alpha}{\alpha}\rangle
    &= \frac{(p-1)q}{l}\Big\{\frac{\|\vec w\|^q}{ q}
    +\langle \vec w, \frac{l}{ (p-1)q}\Bigl(\vec b-\frac{\nabla
          \alpha}{\alpha} \Bigr)\rangle \Big\}\\
    &\geq -\frac{(p-1)q}{ l}\, \frac{l^p}{ [(p-1)q]^p}
    \|\vec b-\frac{\nabla\alpha}{\alpha}\|^p \frac{1}{p}\\
    &=-\frac{l^{p-1}}{p^p}\|\vec  b-\frac{\nabla\alpha}{\alpha}\|^p\\
    &=-\frac{k}{ p^p}\|\vec b-\frac{\nabla\alpha}{\alpha}\|^p
  \end{align*}
  Hence the function $\vec w$ is a solution of the inequality
  \begin{equation*}
    \mathop{\rm div} \vec w+C_2(x)+\frac{p-1}{ l^*}\|\vec w\|^q
    +\langle \vec w, \frac{\nabla\alpha}{\alpha}\rangle \leq 0
  \end{equation*}
  on $\Omega\cap\Omega(a)$. This last inequality is equivalent to
  \begin{equation*}
    \mathop{\rm div}(\alpha \vec w)+\alpha C_2+\frac{p-1}{l^*}\alpha\|\vec w\|^q\leq 0.
  \end{equation*}
  By Theorem \ref{L2} inequality \eqref{eq:P2a} with $C_2$ instead of
  $c$ holds, a contradiction
  to \eqref{eq:P8}. The proof is complete.
\end{proof}

The last theorem makes use of the two-parametric weighting function
$H(t,x)$ from Theorem \ref{L4} to prove the nonexistence of the
solution of Riccati equation.

\begin{theorem}\label{T4}
  Let $\Omega$ be an unbounded domain in $\mathbb{R}^n$ which satisfy
  \textup{(A1)}. Let $H(t,x)$ be the function which
  satisfies hypothesis \textup{(A2)} and has the properties
  \textup{(i)}--\textup{(v)} of Theorem \ref{L4}. If
  \begin{equation}
    \label{eq:limsup...}
    \limsup_{t\to\infty}\Bigl(\int_{S(a_0)}H(t,x)\,\mathrm{d}S\Bigr)^{-1}
    \int_{\Omega(a_0,t)\cap\Omega}\Bigl[H(t,x)c(x)
    -\frac{\|\vec h(t,x)\|^p}{ p^pH^{p-1}(t,x)}\Bigr]\,\mathrm{d}x=\infty,
  \end{equation}
  then equation \eqref{eq:E} is oscillatory in $\Omega$.
\end{theorem}

\begin{proof}
  Suppose that the equation is nonoscillatory. Then the Riccati
  equation \eqref{eq:RicRce} has a solution defined on
  $\Omega\cap\Omega(T)$ for some $T>1$, which is bounded near the
  boundary $\partial\Omega$. Hence \eqref{eq:limsup} of Theorem
  \ref{L4} with $K=p-1$ holds, a contradiction to
  \eqref{eq:limsup...}. Hence the theorem follows.
\end{proof}


\section{Examples}

In the last part of the paper we will illustrate the ideas
from the preceding section.
The specification of the function $\alpha$ in Theorem \ref{T3} leads
to the following oscillation criterion for a conic domain on the
plane. In this case the function $\alpha$ is only the function of a
polar coordinate $\phi$.

\begin{corollary}\label{Co:4.1}
  Let us consider equation \eqref{eq:ENetl} on the
  plane (i.e. $n=2$) with polar coordinates $(r, \phi)$ and let
  \begin{equation}\label{eq:OM}
    \Omega=\{(x,y)\in\mathbb{R}^2: \phi_1< \phi(x,y)<\phi_2\},
  \end{equation}
  where $0\leq\phi_1<\phi_2\leq 2\pi$ and $\phi(x,y)$ is a polar
  coordinate of the point $(x,y)\in\mathbb{R}^2$. Further suppose that the
  smooth function $\alpha\in C^1(\Omega(1),\mathbb{R}^+_0)$ does not depend on
  $r$, i.e.  $\alpha=\alpha(\phi)$. Also, suppose that
  \begin{enumerate}
  \item $\alpha(\phi)\neq0$ if and only if $\phi\in(\phi_1,\phi_2)$
  \item
    \begin{equation*}
%       \label{eq:intI1}
      I_1:=\int_{\phi_1}^{\phi_2}
      \frac{|\alpha'_\phi(\phi)|^p}{4\alpha^{p-1}(\phi)} < \infty,
    \end{equation*}
    where $\alpha'_\phi=\frac{\partial \alpha}{\partial \phi}$.
  \end{enumerate}
  Each one  of the following conditions is sufficient for oscillation of
  \eqref{eq:ENetl} on the domain $\Omega$:
  \begin{enumerate}
  \item $p>2$ and
    \begin{equation}
      \lim_{t\to\infty}
      \int_1^t r\int_{\phi_1}^{\phi_2} c(r,\phi)\,\alpha(\phi)\,\mathrm{d}\phi\,\mathrm{d}r =
      \infty\label{eq:Cor1}
    \end{equation}
  \item $p=2$ and
    \begin{equation}
      \liminf_{t\to\infty}\frac{1}{\ln t}
      \int_1^t r\int_{\phi_1}^{\phi_2} c(r,\phi)\,\alpha(\phi)\,\mathrm{d}\phi\,\mathrm{d}r
      >I_1,\label{eq:Cor2}
    \end{equation}
  \end{enumerate}
  where $c(r,\phi)$ is the potential $c(x)$ transformed into the polar
  coordinates.
\end{corollary}

\begin{proof}
  First let us remind that in the polar coordinates $\,\mathrm{d}x=r\,\mathrm{d}r\,\mathrm{d}\phi$
  and $\,\mathrm{d}S=r\,\mathrm{d}\phi$ holds. Direct computation shows that
  \begin{equation*}
    \int^\infty\Bigl(\int_{\Omega \cap S(t)}\alpha(\phi)\,\mathrm{d}S\Bigr)^{1-q}\,\mathrm{d}t=
    \int_{\phi_1}^{\phi_2}\alpha(\phi)\,\mathrm{d}\phi\cdot\int^\infty t^{1-q}\,\mathrm{d}t.
  \end{equation*}
  and the integral diverges, since $p\geq 2$ is equivalent to $q\leq
  2$. Hence \eqref{eq:P1a} holds.
  Transforming the nabla operator to the polar coordinates gives
  $\nabla \alpha=(0,r^{-1} \alpha'_\phi(\phi))$. Hence, according to
  Theorem \ref{T3}, it is sufficient to show that there exists $k>1$
  such that
  \begin{equation}\label{eq:PrCol11}
    \lim_{t\to\infty} \int_{\Omega\cap\Omega(1,t)}
    \Bigl[c(r,\phi)\alpha(\phi)-\frac k{p^p} \frac{|
    \alpha'_\phi(\phi)|^p}{r^p\alpha^{p-1}(\phi)}\Bigr]\,\mathrm{d}x = \infty.
  \end{equation}
  Since for $p>2$
  \begin{equation*}
    \lim_{t\to\infty}\int_{\Omega\cap\Omega(1,t)}\frac{|\alpha'_\phi(\phi)|^p}
    {r^p\alpha^{p-1}(\phi)}\,\mathrm{d}x
    =
    \int_{\phi_1}^{\phi_2}\frac{|\alpha'_\phi(\phi)|^p}
    {\alpha^{p-1}(\phi)}\,\mathrm{d}\phi\lim_{t\to\infty}    \int_1^t r^{1-p}\,\mathrm{d}r<\infty,
  \end{equation*}
  the conditions \eqref{eq:PrCol11} and \eqref{eq:Cor1} are
  equivalent.

  Finally, suppose $p=2$. From \eqref{eq:Cor2} it follows that there
  exists $t_0>1$ and $\epsilon>0$ such that
  \begin{equation*}
    \frac{1}{\ln
      t}\int_{\Omega\cap \Omega(1,t)}c(r,\phi)\,\alpha(\phi)\,\mathrm{d}x>I_1+2\epsilon
  \end{equation*}
  for all $t\geq t_0$ and hence
  \begin{equation*}
    \int_{\Omega\cap
    \Omega(1,t)}c(r,\phi)\,\alpha(\phi)\,\mathrm{d}x>\Bigl[kI_1+\epsilon\Bigr]\ln t
  \end{equation*}
  where $k=1+\epsilon I_1^{-1}$ holds for $t\geq t_0$.  Since
  \begin{align*}
    kI_1\ln t&=\frac {k  \ln t}4
    \int_{\phi_1}^{\phi_2}{|\alpha'_\phi(\phi)|^2}
    {\alpha^{-1}(\phi)}\,\mathrm{d}\phi\\&=\int_1^t\frac k{4r}\Bigl(\int_{\phi_1}^{\phi_2} {|\alpha'_\phi(\phi)|^2}
    {\alpha^{-1}(\phi)}\,\mathrm{d}\phi\Bigr)\,\mathrm{d}r\\
    &=\int_{\Omega\cap\Omega(1,t)}\frac k{4r^2} {|\alpha'_\phi(\phi)|^2}
    {\alpha^{-1}(\phi)}\,\mathrm{d}x
  \end{align*}
  holds, the last inequality can be written in the form
  \begin{equation*}
    \int_{\Omega\cap \Omega(1,t)}\Bigl[c(r,\phi)\alpha(\phi)-\frac
    k4\,\frac{|\alpha'_\phi(\phi)|^2}{r^2\alpha(\phi)}\Bigr]\,\mathrm{d}x>\epsilon \ln t
  \end{equation*}
  and the limit process $t\to\infty$ shows that \eqref{eq:PrCol11}
  holds also for $p=2$. The proof is complete.
\end{proof}

\begin{example} \rm
  For $n=2$ let us consider the Schr\"odinger equation
  \eqref{eq:Schr}, which in the polar coordinates $(r,\phi)$ reads as
  \begin{equation} \label{eq:SchrPol}
    \frac 1r\frac\partial{\partial r}\Bigl(r\frac{\partial u}{\partial
    r} \Bigr)+\frac 1{r^2}\,\frac{\partial^2u}{\partial \phi^2} +
    c(r,\phi)u=0.
  \end{equation}
  In Corollary \ref{Co:4.1} let us choose $\phi_1=0$, $\phi_2=\pi$,
  $\alpha(\phi)=\sin^2\phi$ for $\phi\in[0,\pi]$ and $\alpha(\phi)=0$
  otherwise. In this case the direct computation shows that the oscillation
  constant $I_1$ in \eqref{eq:Cor2} is $\frac \pi 2$, i.e. the
  equation is oscillatory on the half-plane $\Omega=\{(x_1,
  x_2)\in\mathbb{R}^2:x_2>0\}$ if
  \begin{equation}
    \lim_{t\to\infty}\frac{1}{\ln
        t}\int_{1}^t r\int_0^\pi c(r,\phi)\,\sin^2(\phi)\,\mathrm{d}\phi\,\mathrm{d}r>\frac
        \pi2. \label{eq:ex-rf}
  \end{equation}
  Similarly, the choice $\alpha(\phi)=\sin^3\phi$ gives an oscillation
  constant $3/2$.
\end{example}

\begin{remark} \rm
  It is easy to see that the condition \eqref{eq:ex-rf} can be
  fulfilled also for the function $c$ which satisfy
  $\int_0^{2\pi}c(r,\phi)\,\mathrm{d}\phi=0$ and hence the criteria from
  Theorems \ref{th:JKY} and \ref{th:HW} fails to detect the
  oscillation.
\end{remark}

Another specification of the function $\alpha(x)$ leads to the
following corollary.

\begin{corollary}\label{Co:4.2}
  Let $\Omega$ be an unbounded domain in $\mathbb{R}^2$ specified in Corollary
  \ref{Co:4.1}. Let $A\in C^1([0,2\pi], \mathbb{R}^+_0)$ be a smooth function
  satisfying
  \begin{enumerate}
  \item $A(\phi)\neq 0 $ if and only in $\phi\in(\phi_1,\phi_2)$
  \item $A(0)=A(2\pi)$ and $A'(0+)=A'(2\pi-)$
  \item the following integral converges
    \begin{equation}
      \label{eq:intI2}
      I_2:=\int_{\phi_1}^{\phi_2}\frac{[A^2(\phi)(p-2)^2+(A'(\phi))^2]^{\frac
      p2}}{p^p A^{p-1}(\phi)}\,\mathrm{d}\phi<\infty.
    \end{equation}
  \end{enumerate}
  If
  \begin{equation}
    \label{eq:podm-I2}
    \liminf_{t\to\infty}\frac 1{\ln t}\int_1^t
    r^{p-1}\int_{\phi_1}^{\phi_2} c(r,\phi)A(\phi)\,\mathrm{d}\phi\,\mathrm{d}r>I_2,
  \end{equation}
  then \eqref{eq:ENetl} is oscillatory in $\Omega$.
\end{corollary}

\begin{proof}
  Let $\alpha$ be defined in polar coordinates by the relation
  \begin{equation*}
    \alpha(x(r,\phi))=r^{p-2}A(\phi).
  \end{equation*}
  Computation in the polar coordinates gives
  \begin{align*}
    \int^\infty\Bigl(\int_{\Omega\cap S(t)}\alpha(x)\,\mathrm{d}S\Bigr)^{1-q}\,\mathrm{d}t =&
    \int^\infty
    \Bigl(r^{p-1}\Bigr)^{1-q}\,\mathrm{d}r\int_{\phi_1}^{\phi_2}A(\phi)\,\mathrm{d}\phi \\
    =&\int^\infty\frac1r\,\mathrm{d}r \int_{\phi_1}^{\phi_2}A(\phi)\,\mathrm{d}\phi =
    \infty
  \end{align*}
  and hence \eqref{eq:P1a} holds. The application of the nabla
  operator in polar coordinates yields
  \begin{equation*}
    \nabla\alpha(x(r,\phi))=\Bigl(\frac{\partial\alpha(x(r,\phi))}{\partial r},\frac 1r\,
    \frac{\partial \alpha(x(r,\phi))}{\partial \phi}\Bigr)=r^{p-3}((p-2)A(\phi), A'(\phi))
  \end{equation*}
  and hence on $\Omega$
  \begin{align*}
    \frac{\|\nabla\alpha(x(r,\phi))\|^p}{\alpha^{p-1}(x(r,\phi))}
 &= \frac{r^{p(p-3)}\bigl[(p-2)^2 A^2(\phi)+A'^2(\phi)\bigr]^{p/2}}
    {r^{(p-1)(p-2)}A^{p-1}(\phi)} \\
    &= r^{-2}\frac{\bigl[(p-2)^2 A^2(\phi)+A'^2(\phi)\bigr]^{p/2}}
    {A^{p-1}(\phi)}
  \end{align*}
  holds. Integration over the part $\Omega\cap S(r)$ of the sphere
  $S(r)$ in polar coordinates gives (in view of \eqref{eq:intI2})
  \begin{equation*}
    \int_{\Omega\cap S(r)}\frac{\|\nabla  \alpha(x(r,\phi))\|^p}
    {p^p\alpha^{p-1}(x(r,\phi))}\,\mathrm{d}S=r^{-1} I_2.
  \end{equation*}
  From \eqref{eq:podm-I2} it follows that there exist a real numbers
  $\epsilon>0$ and $t_0>1$ such that
  \begin{equation}\label{...}
    \frac 1{\ln t} \int_1^t r^{p-1} \int_{\phi_1}^{\phi_2}
    c(r,\phi)A(\phi) \,\mathrm{d}\phi\,\mathrm{d}r >I_2+2\epsilon=I_2(1+\epsilon I_2^{-1})+\epsilon
  \end{equation}
  holds for $t>t_0$. Denote $k=1+\epsilon I_2^{-1}$. Clearly $k>1$. From
  \eqref{...} it follows that for $t>t_0$
  \begin{equation*}
    \int_1^t r^{p-1}\int_{\phi_1}^{\phi_2}c(r,\phi)
    A(\phi)\,\mathrm{d}\phi\,\mathrm{d}r>kI_2\ln t +\epsilon\ln t
  \end{equation*}
  holds. This inequality can be written in the form
  \begin{equation*}
    \int_1^t\Bigl[r^{p-1}\int_{\phi_1}^{\phi_2}
    c(r,\phi)A(\phi)\,\mathrm{d}\phi -r^{-1} kI_2\Bigr]\,\mathrm{d}r>\epsilon\ln t
  \end{equation*}
  which is equivalent to
  \begin{equation*}
    \int_{\Omega \cap \Omega(1,t)}\Bigl[c(r,\phi)\alpha(r,\phi) -
    k \frac{\|\nabla \alpha(r,\phi)\|^p}{p^p\alpha^{p-1}(r,\phi)}
    \Bigr] \,\mathrm{d}x >\epsilon \ln t,
  \end{equation*}
  where $\,\mathrm{d}x=r\,\mathrm{d}r\,\mathrm{d}\phi$. Now the limit process $t\to\infty$ shows that
  \eqref{eq:P8} holds and hence \eqref{eq:ENetl} is oscillatory in
  $\Omega$ by Theorem \ref{T3}.
\end{proof}

\begin{example} \rm
  An example of the function $A$ which for $p>1$, $\phi_1=0$
  and $\phi_2=\pi$ satisfies the conditions from Corollary
  \ref{Co:4.2} is $ A(\phi)= \sin^p\phi$ for ${\phi\in(0,\pi)}$ and
  $A(\phi)=0$ otherwise.  In this case the condition
  \begin{multline*}
    \liminf_{t\to\infty}\frac 1{\ln t}\int_1^tr^{p-1}\Big( \int_0^\pi
    c(r,\phi) \sin^p\phi\,\mathrm{d}\phi\Big)\,\mathrm{d}r\\
>\int_0^\pi\frac{\bigl[ (p-2)^2\sin^{2p}\phi
    + p^2\sin^{2p-2}\phi\cos^2\phi
    \bigr]^{p/2}}{p^p\sin^{p(p-1)}\phi}\,\mathrm{d}\phi
  \end{multline*}
  is sufficient for oscillation of \eqref{eq:ENetl} (with $n=2$) over
  the domain $\Omega$ specified in \eqref{eq:OM}. Here $c(r,\phi)$ is
  the potential $c(x)$ transformed into the polar coordinates
  $(r,\phi)$, i.e. $c(r,\phi)=c(x(r,\phi))$.
\end{example}

\begin{corollary}\label{Co:4.3}
  Let us consider the Schr\"odinger equation \eqref{eq:SchrPol} in the
  polar coordinates. Every of the following conditions is sufficient
  for the oscillation of the equation over the half-plane
  \begin{equation}
    \label{eq:OM-Co:4.3}
    \Omega=\{(x_1,x_2)\in\mathbb{R}^2:x_2>0\}.
  \end{equation}
  \begin{enumerate}
  \item There exists $\lambda>1$ such that
    \begin{equation}
      \label{eq:pod-Co4.3-1}
      \limsup_{t\to\infty} t^{-\lambda} \int_1^t
      (t-r)^\lambda\Bigl(r\int_0^\pi c(r,\phi)\sin^2\phi\,\mathrm{d}\phi-\frac
      \pi{2r} \Bigr)\,\mathrm{d}r=\infty.
    \end{equation}
  \item There exists $\lambda>1$ and $\gamma<0$ such that
    \begin{equation}
      \label{eq:pod-Co4.3-2}
      \limsup_{t\to\infty} t^{-\lambda} \int_1^t r^{\gamma+1}
      (t-r)^\lambda \int_0^\pi c(r,\phi)\sin^2\phi\,\mathrm{d}\phi\,\mathrm{d}r=\infty.
    \end{equation}
  \end{enumerate}
\end{corollary}

\begin{proof}
  For $\gamma\leq 0$ let us define
  \begin{equation*}H(t,x)=
    \begin{cases}
      r^\gamma (t-r)^\lambda\sin^2\phi &\phi\in (0\pi)\\
      0&\text{otherwise,}
    \end{cases}
  \end{equation*}
  where $(r,\phi)$ are the polar coordinates of the point
  $x\in\mathbb{R}^2$. In the polar coordinates
  $\nabla=(\frac\partial{\partial r}, \frac 1r\,\frac
  \partial{\partial\phi})$. Hence
  \begin{align*}
    \vec h(t,x(r,\phi))&=-\nabla H(t,x(r,\phi))\\
    &=-\Bigl(r^{\gamma-1}(t-r)^{\lambda-1}(\gamma (t-r)-\lambda
    r)\sin^2\phi ,\ 2r^{\gamma-1}(t-r)^\lambda\sin\phi\cos\phi\Bigr)
  \end{align*}
  and consequently
  \begin{align}
    \frac{\|\vec h(t,x(r,\phi))\|^2}{H(t,x(r,\phi))}
&=\gamma^2r^{\gamma-2}(t-r)^\lambda\sin^2\phi-2\lambda\gamma
    r^{\gamma-1} (t-r)^{\lambda-1}\sin^2\phi\nonumber\\
&\quad +\lambda^2r^\gamma (t-r)^{\lambda-2}\sin^2\phi+4r^{\gamma-2}
    (t-r)^\lambda \cos^2\phi.\label{eq:hH}
  \end{align}
  Now it is clear that for $\lambda>1$ inequality $\lambda-2>-1$ holds.
  Hence the integral over $\Omega\cap\Omega(1,t)$ converges and
  \eqref{eq:konv-h} for $p=2$ holds. Further
  \begin{equation*}
    \int_{S(r)\cap \Omega}H(t,x)\,\mathrm{d}S=r\int_0^\pi r^\gamma(t-r)^\lambda
    \sin^2\phi\,\mathrm{d}\phi =\frac\pi 2 r^{\gamma+1}(t-r)^\lambda
  \end{equation*}
  and the condition (v) of Theorem \ref{L4} holds with
  $k(r)=r^{-1-\gamma}$. It remains to prove that the conditions
  \eqref{eq:pod-Co4.3-1} and \eqref{eq:pod-Co4.3-2} imply
  the condition \eqref{eq:limsup...}. Since
  $\int_0^\pi\sin^2\phi\,\mathrm{d}\phi=\int_0^\pi \cos^2\phi\,\mathrm{d}\phi=\frac\pi2$,
  it follows from \eqref{eq:hH} that
  \begin{align}
    \int_{S(r)\cap \Omega} \frac{\|\vec h(t,x(r,\phi))\|^2}
    {H(t,x(r,\phi))}\,\mathrm{d}S =&
    \frac\pi2(\gamma^2+4)r^{\gamma-1}(t-r)^\lambda-
    \pi\lambda\gamma  r^{\gamma} (t-r)^{\lambda-1}\nonumber\\
    &+\frac\pi2\lambda^2r^{\gamma+1} (t-r)^{\lambda-2}. \label{al:konv}
  \end{align}
  Next we will show that
  \begin{align}
    \lim_{t\to\infty}t^{-\lambda}\int_1^t
    r^\gamma(t-r)^{\lambda-1}\,\mathrm{d}r&<\infty \label{al:1}\\
    \lim_{t\to\infty}t^{-\lambda}\int_1^t
    r^{\gamma+1}(t-r)^{\lambda-2}\,\mathrm{d}r&<\infty \label{al:2}
    \intertext{and for $\gamma<0$ also}
    \lim_{t\to\infty}t^{-\lambda}\int_1^t
    r^{\gamma-1}(t-r)^{\lambda}\,\mathrm{d}r&<\infty \label{al:3}
   \end{align}
  holds. Inequality \eqref{al:1} follows from the estimate
  \begin{equation*}
    \int_1^t r^\gamma(t-r)^{\lambda-1}\,\mathrm{d}r\leq\int_1^t 1^\gamma
    (t-r)^{\lambda-1} \,\mathrm{d}r=\frac 1\lambda (t-1)^\lambda.
  \end{equation*}
  Integration  by parts shows
  \begin{equation*}
    \int_1^tr^{\gamma+1}(t-r)^{\lambda-2}\,\mathrm{d}r=
    \frac{(t-1)^{\lambda-1}}{\lambda-1}  +\frac{\gamma+1}{\lambda-1}
    \int_1^t r^\gamma(t-r)^{\lambda-1}\,\mathrm{d}r
  \end{equation*}
  and in view of \eqref{al:1} inequality \eqref{al:2} holds as
  well. Finally, for $\gamma<0$ integration by parts gives
  \begin{equation*}
    \int_1^t
    r^{\gamma-1}(t-r)^{\lambda}\,\mathrm{d}r=\frac{(t-1)^\lambda}{\gamma} +
    \frac \lambda\gamma\int_1^t r^\gamma(t-r)^{\lambda-1}\,\mathrm{d}r
  \end{equation*}
  and again the inequality \eqref{al:3} follows from \eqref{al:1}.
  Hence the terms from \eqref{al:konv} have no influence on the
  divergence of \eqref{eq:limsup...} (except the term
  $r^{-1}(t-r)^\lambda$ which appears for $\gamma=0$) and hence \eqref{eq:limsup...} follows from
  \eqref{eq:pod-Co4.3-1} and \eqref{eq:pod-Co4.3-2}, respectively.
  Consequently, the equation is oscillatory by Theorem \ref{T4}.
\end{proof}


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\end{document}




% LocalWords:  Picone Schr odinger majorant homogenity semilinear Nehari Hile
% LocalWords:  nonoscillation Kandelaki Schminke Wintner Atakarryev Kneser
% LocalWords:  Toraev superlinear