\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 78, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/78\hfil Estimates for critical groups]
{Estimates for critical groups of solutions to quasilinear
elliptic systems}

\author[J. Carmona, S. Cingolani, \& G. Vannella\hfil EJDE--2003/78\hfilneg]
{Jos\'e Carmona, Silvia Cingolani, \& Giuseppina Vannella}

\address{Jos\'e Carmona \newline
Department {\'A}lgebra y An{\'a}lisis Matem{\'a}tico
Universidad de Almer\'{\i}a\\
Almer\'{\i}a, Spain} 
\email{jcarmona@ual.es}

\address{Silvia Cingolani \newline
Dipartimento di Matematica,
Politecnico di Bari \\
Bari, Italy} 
\email{cingolan@dm.uniba.it}

\address{Giuseppina Vannella  \newline
Dipartimento di Matematica,
Politecnico di Bari \\
Bari, Italy} 
\email{vannella@dm.uniba.it}

\date{}
\thanks{Submitted January 15, 2003. Published July 28, 2003.}
\subjclass[2000]{58E60, 35J60, 35B65}
\keywords{Quasilinear elliptic systems, critical groups, Morse index}


\begin{abstract}
 In this work we study a class of functionals, defined on Banach spaces,
 associated with quasilinear elliptic systems. Firstly, we prove some
 regularity results about the critical points of such functionals and
 then we estimate the critical groups in each critical point via its
 Morse index.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{Lemma}[theorem]{Lemma}


\section{Introduction and statement of the results}

Let  $X$ be a Banach space and $f:X \to \mathbb{R}$ a
$C^2$ functional. For any $a\in \mathbb{R}$, we denote by $f^a$ the
sublevel $ \{ v \in X : f(v) \leq a \}$. Let $u$ be a critical
point of $f$, at level $c=f(u)$. We call
\[
C_{q}( f,u) = H^{q}( f^{c},f^{c}\setminus \{u\})
\]
the q-th critical group of $f$ at $u$, $q=0,1,2,..,$ where
$H^q(A,B)$ stands for the q-th Alexander-Spanier cohomology group
of the pair $(A,B)$ with coefficients in  ${\mathbb{K}}$ \cite{3}.


 If $X$ is a Hilbert space and $f$ is a $C^2$ Euler
functional on $X$, which satisfies some {\sl apriori} compactness
conditions on the sublevels, the changes of topology of its sublevels
 can be checked by computing the {\sl critical groups}
of the functional in the critical points. The estimates of the critical
groups in a critical point become a quite clear fact if the critical point is
{\sl non-degenerate}, namely the second derivative of the functional in the
critical point is an isomorphism. A classical result, based on Morse Lemma,
allows us to relate the critical groups in the non-degenerate critical point
$u$ to its Morse index, namely the supremum of the dimensions in which the
second derivative
 $f''(u)$ of $f$ in $u$ is negative definite.
For reader's convenience, we recall the following theorem (see
also \cite{2}).

\begin{theorem}\label{impo}
Suppose $H$ a Hilbert space and $I\in C^2(H, \mathbb{R})$. Let $u$ be a
non-degenerate critical point of $I$ with Morse index $m$. Then
$$
C_q(I,u)\cong {\mathbb{K}}\quad\text{if $q=m$, and}\quad
C_q(I,u)= \{ 0\} \text{ if ${q \ne m}$.}
$$
 Nevertheless, if $m=+\infty$, we always have $ C_q(I,u)= \{ 0 \}$.
\end{theorem}

Gromoll and Meyer extended the previous theorem for the case of an isolated
critical point $u$, possibly degenerate, when the second derivative of the
functional in the critical point $u$ is a Fredholm operator with zero index.
These extensions are based on generalizations of the Morse splitting lemma
(see \cite{2}).

 In a Banach space framework, the estimate of the critical groups is
a very interesting, but not classical fact. A lot of problems arise when
trying to develop a local Morse theory, as Morse splitting type lemma does
not hold and it is not {\sl a priori} clear what can be a reasonable
definition of non-degenerate critical point in a Banach space, which is not
isomorphic to its dual space. Moreover, some crucial ingredients in Morse
theory for Hilbert spaces, like the Fredholm properties of the second
derivative of the functional in the critical points, lack in a Banach space
setting.

In recent papers by Cingolani and Vannella \cite{CV1,CV}, the
authors give a connection between critical groups of a solution to
a quasilinear equation, involving $p$-Laplacian, and its Morse
index. In \cite{CV}, a new definition of non-degenerate critical
point (see also \cite{2,9,10} for different definitions) is
introduced involving only the injectivity of the second derivative
of the Euler functional in the critical point. Such weak
nondegeneracy is enough to regain a suitable splitting of the
Banach space and to evaluate the critical groups of the critical
points via Morse index. See also \cite{CV2} for Marino-Prodi
perturbation type results \cite{MP}, using the new definition of
nondegeneracy given in \cite{CV}.

In this paper we aim to extend the results in \cite{CV} for a
class of quasilinear elliptic systems.
We deal with functionals $I: W^{1,p}_0(\Omega,\mathbb{R}^m) \to \mathbb{R}$ defined by
\begin{equation*}%\label{func}
I(u) = \int_\Omega \Big( {\frac{1}{p}} |\nabla u |^p +
{\frac{1}{2}} |\nabla u|^2 + g(u) \Big) dx, \quad
 u\in W_0^{1,p}(\Omega,\mathbb{R}^m),
\end{equation*}
where $\Omega $ is a bounded domain of $\mathbb{R}^n$ with
smooth boundary $\partial \Omega $, $n \ge 2$, $m \ge 2$, $p \geq 2$
and, as usual
$$
|\nabla u|^{p} =
\Big(\sum_{i=1}^{m}\sum_{\alpha=1}^{n} \big(\frac{\partial u^i}{\partial
 x_\alpha}\big)^2
\Big)^{p/2}.
$$
We assume that $g \in C^2(\mathbb{R}^{m}, \mathbb{R})$ and that
\begin{itemize}
\item[(G1)] For any $\xi \in \mathbb{R}^m$, $|g''(\xi)| \leq c_1
|\xi|^q + c_2$  with $c_1,c_2$ positive constants and \newline
$0\leq q < {\frac{np}{(n-p)}}-2$ if $n >p$, while $q$ is any
positive number, if $n=p$.
\end{itemize}
 If $n<p$, no restrictive assumption is required. \smallskip

 The functional $I$ is $C^2$ on $W_0^{1,p}(\Omega, \mathbb{R}^m)$
and critical points of $I$ correspond to weak solutions of the
quasilinear elliptic system
\begin{equation} \label{P}
\begin{gathered}
- \Delta u - \Delta _{p}u + g'(u) =0 \quad \text{on } \Omega \\
u=0 \quad \text{on } \partial \Omega
\end{gathered}
\end{equation}
 where, as usual
 $\Delta_p u = (\mathop{\rm div} (|\nabla u|^{p-2} \nabla u^i))_{i=1}^m$
for any $p \geq 2$, and $\Delta u=\Delta_2 u$.

 We underline that the variational setting for problem \eqref{P} is the
Banach space $W^{1,p}_0(\Omega,\mathbb{R}^m)$, which cannot be
equipped by an equivalent Hilbert norm if $p\ne 2$. In this case
we also note that:
\begin{itemize}
\item Any critical point of $I$ is degenerate in the classical sense used
in Hilbert space, in fact $I^{\prime \prime }\left( u\right)
:W_{0}^{1,p}\left( \Omega ,\mathbb{R}^m \right) \rightarrow
W^{-1,p^{\prime }}\left( \Omega ,\mathbb{R}^m \right) $ ($1 /{p'} +1/p
=1$) is not invertible, since $W_{0}^{1,p}\left( \Omega ,\mathbb{R}^m
\right) $ is not isomorphic to its dual;
\item  $I^{\prime \prime }\left( u\right) $ cannot be a
Fredholm operator. Hence extensions of Morse's lemma such as that of Gromoll
and Meyer type cannot be applied.
\end{itemize}


In spite of the above difficulties, we are able to relate critical groups to
differential properties of the critical points of $I$, like in the Hilbert
space case.  Before stating the main results, let us denote by
$m(I,u)$ the Morse index of $I$ at $u$ and by $m^*(I,u)$ the sum of $m(I,u)$
and the dimension of the kernel of $I''(u)$ in $W_0^{1,p}(\Omega,\mathbb{R}^m)$.

\begin{theorem}\label{generalizzato}
 Let $u$ be a critical point of $I$ such that $I''(u)$ is
injective. Then $m(I, u)$ is finite and
\begin{gather*}
 C_q(I, u) \cong {\mathbb{K}}, \quad \hbox{if }  q = m(I,u),\\
 C_q(I,u) = \{ 0 \}, \quad \hbox{if }  q \ne m(I, u).
\end{gather*}
\end{theorem}

 The main idea in the proof of Theorem \ref{generalizzato} is the
introduction of a Hilbert space $H$ which is obtained as closure of the space
$C^{\infty}_0(\Omega,\mathbb{R}^m)$ with respect to a scalar product depending on the
critical point $u \in W^{1,p}_0(\Omega, \mathbb{R}^m)$. We remark that the Banach
space $W^{1,p}_0(\Omega, \mathbb{R}^m)$ is continuously embedded in such an auxiliary
Hilbert space. To this aim it is crucial to prove some regularity results
concerning the weak solutions to problem \eqref{P} (see Section 2). Furthermore
it is possible to extend the second derivative of the Euler functional $I$ at
the critical point $u$ to a Fredholm operator with zero index from $H$ to its
dual space and so to obtain a suitable splitting of the variational space
$W_0^{1,p}(\Omega, \mathbb{R}^m)$.

Finally, by using some ideas recently developed in \cite{CV}, we
are able to obtain a finite dimensional reduction and to prove
that the critical groups of $I$ in $u$ are isomorphic to the
critical groups of a suitable function defined on a finite
dimensional subspace of $W_0^{1,p}(\Omega, \mathbb{R}^m)$.

 It is important to note that when Theorem~\ref{generalizzato} is
applied to problems like \eqref{P}, the Morse series is not yet formal, but it is
a real equality between polynomials.

\smallskip

 In the case of isolated, possibly degenerate, critical points we
also prove the following statement.

\begin{theorem}\label{indicelargo}
 Let $u$ be an isolated critical point of $I$. Then $m(I, u)$
and $m^*(I,u)$ are finite and  we have
\[
C_q(I,u) = \{ 0 \},  \quad \hbox{if } q < m(I,u) \quad \hbox{or } q > m^*(I, u).
\]
\end{theorem}

\subsection*{Notation}
We denote by $(\cdot | \cdot)$ the usual scalar product
in $\mathbb{R}^n$.

$L^{p}(\Omega)$ denotes the usual Lebesgue space with norm
$\left(\int_\Omega |v|^p\right)^{1/p}$. In the vector valued space
$L^{p}(\Omega,\mathbb{R}^m)=L^{p}(\Omega)\times \dots\times
L^{p}(\Omega)=\left(L^{p}(\Omega)\right)^m$ we consider the norm
$\|u\|_{p}^p= \int_\Omega |u|^p$.

$W^{1,p}_0(\Omega)$ denotes the usual Sobolev
space with norm $\left(\int_\Omega |\nabla v|^p\right)^{1/p}$. In
the vector valued space
$W^{1,p}_0(\Omega,\mathbb{R}^m)=W^{1,p}_0(\Omega)\times \dots\times
W^{1,p}_0(\Omega)=\big(W^{1,p}_0(\Omega)\big)^m$ we consider
the norm $\|u\|=\|\nabla u \|_{p}$. For any $p<n$ the critical
Sobolev exponent will be denoted by $p^*=\frac{pn}{n-p}$.

 $W^{-1,p^{\prime}}(\Omega,\mathbb{R}^m)$ denotes the dual space of
$W^{1,p}(\Omega,\mathbb{R}^m)$, and $1 /{p'} +1/p =1$. Moreover we denote
by $\langle \cdot , \cdot \rangle : W^{-1,p'}(\Omega,\mathbb{R}^m) \times
W_0^{1,p}(\Omega,\mathbb{R}^m) \to \mathbb{R}$ the duality pairing.

 $B_r(u)= \{ v \in W^{1,p}_0(\Omega, \mathbb{R}^m) : \| v - u\| < r
\}$, where $u \in W^{1,p}_0(\Omega,\mathbb{R}^m)$ and $r>0$.

 $|A|$ and $\mathop{\rm meas} A$ denotes the Lebesgue measure of $A\subset
 \mathbb{R}^n$.



\section{Regularity}

 In this section,  using the interior regularity result contained in
\cite{tolksdorfs} and an argument similar to those contained in
\cite{egnell,tolksdorf} for the scalar case, we prove that every solution
$u\in W_0^{1,p}(\Omega,\mathbb{R}^m)$ of \eqref{P} also belongs to $C^{1}(\overline\Omega,\mathbb{R}^m)$.

We recall here the Stampacchia lemma \cite[Lemma 4.1]{stam}, which will be
the main tool in proving that $u\in L^{\infty}(\Omega,\mathbb{R}^m)$.

\begin{Lemma}\label{stampacchia}
Let $\varphi:[0,+\infty[\to \mathbb{R}$ be a non negative, non increasing real valued
function. If there exists positive constants $C,\alpha, \beta$ such that
$$
\varphi(h)\leq\frac{C}{(h-k)^\alpha}\varphi(k)^\beta, \, h>k\geq 0,
$$
then
\begin{itemize}
\item[{\rm (i)}] if $\beta>1$ then $\varphi(d)=0$ with
$d^\alpha=C\varphi(0)^{\beta-1}2^{\alpha\beta/(\beta-1)}$
\item[{\rm (ii)}] if $\beta=1$ then $\varphi(h)\leq \varphi(0)
\exp{(1-\frac{h}{(eC)^{(1/\alpha)}})}$
\item[{\rm (iii)}] if $\beta<1$ then $\varphi(h)\leq \frac{
2^{\frac{\alpha}{(1-\beta)^2}} [C^{\frac{1}{1-\beta}}+
2^{\frac{\alpha}{1-\beta}}
\varphi(1)]}{h^{\frac{\alpha}{1-\beta}}},\:h>1.$
\end{itemize}
\end{Lemma}

Next we prove the main results for this section.

\begin{Lemma}\label{linfty}
If $u\in W_0^{1,p}(\Omega,\mathbb{R}^m)$ is a solution to \eqref{P} then $L^\infty(\Omega)$. Moreover there
exists $M>0$ such that $\|u\|_{L^\infty} \leq M$.
\end{Lemma}

\begin{proof}
During this proof $c,c_i$ will denote positive constants
independent on $u$ that may change between consecutive steps.
 We focus on the case $p\leq n$. In the other case the
proof follows directly by the Sobolev embedding.

  Firstly, we note that condition (G1) implies
in particular that
\begin{enumerate}
\item[(G2)]
$|g'(\xi)| \leq c_1 |\xi|^s + c_2$  with $c_1,c_2$ positive
constants and $1 \leq s < p^*-1$ if $n >p$, while $s\geq 1$, if
$n=p$.
\end{enumerate}
We consider now for every $k\in\mathbb{R}^+$ the function
$G_k:\mathbb{R}\to \mathbb{R}$ given by
$$
G_k(s)= \begin{cases}
s+k & s\leq -k,\\
0 & -k<s \leq k,\\
s-k & s>k\,.
\end{cases}
$$
Thus, for every $j=1,\dots,m$ we can take
$\phi_j=(\delta_{ij}G_k(u^j))_{i=1,\dots,m}$ as test function in
the weak equation satisfied by $u$ and using $(G_2)$ we have
\begin{equation}\label{emis}
\|\nabla G_k(u^j)\|_p^p  \leq  \int_\Omega (1+|\nabla
u|^{p-2})\nabla u \nabla (\delta_{ij}G_k(u^j)) \leq \int
\limits_{\Omega_k} (c_1|u|^{s}+c_2)|G_k(u^j)|,
\end{equation}
where $\Omega_k\equiv\{x\in \Omega:\, |u^j(x)|>k\}$.

 We first consider the case $p<n$. Let
$r>s\frac{p^*}{p^*-1}$ be such that $u \in L^r(\Omega)$. Writing
$|v|^s=c_1|u|^s+c_2$ and using the Sobolev and H\"older
inequalities, by (\ref{emis}) we yield,
\begin{equation}\label{tom}
\|G_k(u^j)\|_{p^*}^p\leq c \|v\|_r^s \|G_k(u^j)\|_{p^*}(\mathop{\rm
meas }\Omega_k)^{(1-s/r-1/p^*)}.
\end{equation}
 Taking into account that, for every $h> k$,
$|G_k(u^j)|\geq h-k$ in $\Omega_h$, (\ref{tom}) implies
$$
(h-k)^{p-1}(\mathop{\rm meas }\Omega_h)^{(p-1)/p^*}\leq  c \|v\|_r^s
(\mathop{\rm meas }\Omega_k)^{(1-s/r-1/p^*)},
$$
or equivalently
\begin{equation}
\mathop{\rm meas}\Omega_h\leq \frac{ c \|v\|_r^{\frac{sp^*}{p-1}} (\mathop{\rm
meas }\Omega_k)^{\frac{(p^*-1-sp^*/r)}{p-1}}}{(h-k)^{p^*}}.
\end{equation}
Now we apply Stampacchia's Lemma  with $\varphi(h)=\mathop{\rm meas}\Omega_h$,
$C=c\|v\|_r^{\frac{sp^*}{p-1}}$, $\alpha=p^*$ and
$\beta=\frac{(p^*-1-sp^*/r)}{p-1}$, to prove that:
\begin{enumerate}
\item[{\rm (i)}] if
$u \in L^r(\Omega)$ with $r>\frac{sp^*}{p^*-p}=\frac{sn}p$,
 then $u^j\in L^\infty(\Omega)$ and $\|u^j\|_\infty \leq
c\|v\|_r^{s/(p-1)}$,
\item[{\rm (ii)}] if
$u \in L^r(\Omega)$ with $r=\frac{sn}p$, then $u^j\in L^t(\Omega)$ for
$t\in[1,\infty)$ and $\|u^j\|^t_t\leq c + c'\|v\|_r^{ts/(p-1)}$,
\item[{\rm (iii)}] if
$u \in L^r(\Omega)$ with $r<\frac{sn}p$, then $u^j\in L^t(\Omega)$ for
$t=\frac{p^*r(p-1)}{(p-p^*)r+p^*s}-\delta$ and $\delta>0$
arbitrarily small. Moreover, $\|u^j\|^t_t\leq c +
c'\|v\|_r^{(t+\delta)s/(p-1)}$.
\end{enumerate}
Item {\rm (i)} follows from the fact that $\mathop{\rm meas }\Omega_d=0$. Hence
\[
u^j\leq
d=(C\varphi(0)^{\beta-1}2^{\alpha\beta/(\beta-1)})^{1/\alpha}=
c_1\|v\|_r^{\frac{s}{p-1}}.
\]
To prove items {\rm (ii) and (iii)} we take
\[
T_h(s)=\begin{cases}
-h & s\leq -h, \\
s & -h<s \leq h,\\
h & s>h.
\end{cases}
\]
It is clear that $T_h(u^j)\to u^j$ a.e. in $\Omega$. Now we shall apply the
Vitali Theorem (cf. \cite{HS}) to prove that $u^j \in L^t(\Omega)$ and  the
convergence is strong in $L^t(\Omega)$.

 Firstly by Stampacchia Lemma we deduce that
\begin{equation}\label{sta}
h^t (\mathop{\rm meas}\Omega_h)\to 0
\end{equation}
for any $t>1$ in the case {\rm ii)} and for
$t<\frac{\alpha}{1-\beta}=\frac{p^*r(p-1)}{(p-p^*)r+p^*s}$ in the
case {\rm iii)}. We note that, in both cases, it is possible to
choose $t>1$.

 To apply Vitali Theorem, we need to prove that
for any $E \subset \Omega$,
 \[
\lim_{|E|\to 0}\int_{E}  |T_n(u^j)|^t=0,
 \]
uniformly in $n$. A sufficient condition is
 to show that
 \begin{equation} \label{nk}
\lim_{k\to +\infty}\int_{B_{n,k}\equiv \{|T_n|\geq k\}}
 |T_n(u^j)|^t=0,
 \end{equation}
 uniformly in $n$. First we observe that $B_{n,k}=\emptyset$ if
$k>n$ and $B_{n,k} = B_{k,k}=\Omega_k$ for every $k\leq n$.
Moreover, if $n\geq k$, we can write
\begin{align*}
\int_{B_{n,k}} |T_n(u^j)|^t &= \int_{\Omega_k}|T_n(u^j)|^t\\
&= \sum_{s=k}^{n-1} \int_{\Omega_{s}\setminus \Omega_{s+1}}
|T_n(u^j)|^t + \int_{\Omega_{n}} |T_n(u^j)|^t \\
& \leq  \sum_{s=k}^{n-1} (s+1)^t
\left(|\Omega_{s}|-|\Omega_{s+1}|\right)+n^t|\Omega_n|.
\end{align*}
Since (\ref{sta}) holds, to prove (\ref{nk}) uniformly in
$n$, it is sufficient to show the convergence of
$$
\sum (n+1)^t \left(|\Omega_{n}|-|\Omega_{n+1}|\right).
$$
To this aim, it is sufficient to prove the convergence of the two
series
$$
\sum \left( (n+1)^t |\Omega_{n}|-n^t|\Omega_{n}|\right) \quad
\hbox{ and } \quad \sum \left( n^t
|\Omega_{n}|-(n+1)^t|\Omega_{n+1}|\right).
$$
The first series converges as, by Stampacchia lemma, we can observe
that
$$
\left((n+1)^t -n^t\right)|\Omega_{n}|\leq
t(n+1)^{t-1}|\Omega_{n}|\leq
C_t\big(\frac{n+1}{n}\big)^{t-1}\frac{1}{n^{1+\gamma}}
$$
for some constant $C_t\in \mathbb{R}$ and
$0<\gamma=\frac{\alpha}{1-\beta}-t$.
Also the second series is convergent, due to
(\ref{sta}).
As a consequence we have that
\begin{itemize}
\item If $r=\frac{sn}p$ then $u^j\in L^t(\Omega)$
for $t\in[1,\infty)$
\item If $r<\frac{sn}p$ then $u^j\in L^t(\Omega)$ for
$t=\frac{p^*r(p-1)}{(p-p^*)r+p^*s}-\delta$ and $\delta>0$
arbitrarily small.
\end{itemize}
Finally, the estimates on the norms follows from the convexity of
the real function $s^t$ with $t>1$ and the estimates of
$\varphi(h)$ in Lemma $\ref{stampacchia}$. More precisely,
$$
\|u^j\|_t^t \leq (\|u^j-T_h(u^j)\|_t+\|T_h(u^j)\|_t)^t\leq
c_1+c_2\|T_h(u^j)\|_t^t\leq c_1+c_3h^t,
$$
for each fixed $h$ big enough. Hence by Lemma $\ref{stampacchia}$
we have, in the case {\rm (ii)}, that
$$
h^t\leq (\log{e\varphi(0)/\varphi(h)})^t(e c)^{t/p^*}\|v\|_r^{st/(p-1)}
$$
and in the case {\rm (iii)}
$$ h^{t+\delta}\leq \frac{c_4
\|v\|_r^{\frac{sp^*r}{r(p-p^*)+sp^*}}+ c_5\varphi(1)}{\varphi(h)},
$$
which concludes the estimates in items {\rm (ii)} and {\rm (iii)}
above. We observe that this argument can be done for every
$j=1,\dots,m$, and hence those estimates above, remain valid if we
replace $u^j$ by $u$.

 Since $u\in L^{p^*}(\Omega,\mathbb{R}^m)$ and
$p^*>s\frac{p^*}{p^*-1}$, we can argue as before for $r_0=p^*$.
Thus, if $p^*>\frac{sn}p$ we conclude by item {\rm i)}. In the
case $p^*=\frac{sn}p$ we use item {\rm ii)} in order to take
$r_1>\frac{sn}p$ and conclude again by item {\rm i)}. Finally, in
the case $p^*<\frac{sn}p$ we can take
$$
r_1=\frac{p^*r_0(p-1)}{(p-p^*)r_0+p^*s}-\delta_1>r_0.
$$
As before, if $r_1\geq \frac{sn}p$ we easily conclude. In
other case we take
$$
r_2=\frac{p^*r_1(p-1)}{(p-p^*)r_1+p^*s}- \delta_2.
$$
By an iterative argument we conclude after a finite number of
steps. Indeed, in other case, we have that $r_n$ is bounded, where
$r_n$ is defined recurrently by
\begin{gather*}
r_0=p^* \\
r_{n+1}=\frac{p^*r_n (p-1)}{(p-p^*)r_n+p^*s}-\delta_{n+1}\,,
\end{gather*}
where $\lim \delta_n=0$. Moreover, $r_n$ is non decreasing and so
it converges to $r\in(p^*,\frac{sn}p]$ that satisfies
$$
r=\frac{p^*r(p-1)}{(p-p^*)r+p^*s},
$$
that is, $p^*(p-1)=(p-p^*)r+p^*s$, which implies that
$r=\frac{p^*(p-1-s)}{p-p^*}<p^*$ and
this is a contradiction.

 In the case $p=n$ we can choose $q>p$ and
$r>\frac{qs}{q-p}$ and argue as before with $p^*$ replaced by $q$.
In this case we finish by item {\rm i)} in the Stampacchia Lemma.
\end{proof}

\begin{theorem}\label{bin}
If $u$ is a solution to \eqref{P}, then $u\in C^{1}(\overline\Omega,\mathbb{R}^m)$.
\end{theorem}

\begin{proof} Let $u_0\in W_0^{1,p}(\Omega,\mathbb{R}^m)$ be a solution of \eqref{P}. We consider the
problem
\begin{equation}  \label{sistemac}
\begin{gathered}
-\Delta u -\Delta_p u + g'(u_0) =  0, \quad x\in \Omega, \\
u  =  0, \quad  x\in \partial \Omega,
\end{gathered}
\end{equation}
and we will prove that every $L^{\infty}(\Omega,\mathbb{R}^m)$-solution of (\ref{sistemac})
(by Lemma \ref{linfty}, in particular $u_0$) is
$C^{1}(\overline\Omega,\mathbb{R}^m)$-regular. Indeed, the interior
regularity follows immediately from the estimates in
\cite{tolksdorfs}. Let us show how to obtain the regularity up to
the boundary. Since $\partial \Omega$ is smooth, we can assume
without lost of generality that, near each fixed $x^0\in\partial
\Omega$,
\begin{equation*} %\label{front}
x \in \Omega \; \iff \; x=(x',x_n)\in \mathbb{R}^{n-1}\times \mathbb{R}
\hbox{ and }   x_n>a(x')
\end{equation*}
$a:\mathbb{R}^{n-1}\to \mathbb{R}$ being a $C^2$-function. Thus, the change of
variables $y(x_1,\dots,x_n)$ given by
\begin{align*}
&y_1=x_1-x^0_1,\\
&y_2=x_2-x^0_2,\\
&\dots.\\
&y_{n-1}=x_{n-1}-x^0_{n-1},\\
&y_n=x_n-a(x_1,\dots,x_{n-1}),
\end{align*}
is an invertible map between $\Omega\cap U$ and
$D=\{y=(y_1,\dots,y_n):\, |y|<\delta,\, y_n>0\}$, where $\delta>0$
and $U$ neighborhood of $x_0$ are suitably chosen.

 Let us note that, given $\phi\in C_0^{1}(D,\mathbb{R}^m)$, then
$\phi(y^{-1})\in C_0^{1}(\Omega\cap U,\mathbb{R}^m)$. This leads, for any
solution $u\in W_0^{1,p}(\Omega,\mathbb{R}^m)\cap L^{\infty}(\Omega,\mathbb{R}^m)$ of (\ref{sistemac}), to the
integral equality
\begin{multline} \label{ecui}
 \int_{\Omega\cap U} \Big(1+
\Big|\sum_{i=1}^{m}\sum_{j=1}^{n} \frac{\partial u^i}{\partial
x_j}\frac{\partial u^i}{\partial x_j} \Big|^{\frac{p-2}2}\Big)
\sum_{i=1}^{m}\sum_{j=1}^{n}\frac{\partial u^i}{\partial x_j}
\frac{\partial \phi^i(y^{-1})}{\partial x_j} \, dx \\
+ \int_{\Omega\cap U} \sum_{i=1}^{m} g_i'(u_0)\phi^i(y^{-1}) \,
dx =0.
\end{multline}
 We perform the change of variables in (\ref{ecui}) and
taking into account that $|\frac{d x}{d y}|=1$, we obtain
\begin{multline*}
 \int_{D} \Big(1+\Big|\sum_{i=1}^{m}\sum_{j,l,k=1}^{n}
\frac{\partial y_l}{\partial x_j}\frac{\partial y_k}{\partial x_j}
\frac{\partial u^i}{\partial y_l}\frac{\partial u^i}{\partial
y_k}\Big|^{\frac{p-2}2}\Big)
\sum_{i=1}^{m}\sum_{j,l,k=1}^{n}\frac{\partial y_l}{\partial
x_j}\frac{\partial y_k}{\partial x_j} \frac{\partial u^i}{\partial
y_l}\frac{\partial \phi^i}{\partial y_k} \, dy
\\
 + \int_{D} \sum_{i=1}^{m}
g_i'(u_0)\phi^i  \, dy =0,
\end{multline*}
for any $\phi\in C_0^{1}(D,\mathbb{R}^m)$, where we have denoted by $u(y)$
and $g'(u_0(y))$, the functions
\begin{gather*}
u(y_1+x^0_1,\dots,y_{n-1}+x^0_{n-1},y_n+a(y_1+x^0_1,\dots,y_{n-1}+x^0_{n-1})),\\
g'(u_0(y_1+x^0_1,\dots,y_{n-1}+x^0_{n-1},y_n+a(y_1+x^0_1,\dots,y_{n-1}+x^0_{n-1}))).
\end{gather*}
Now we extend $u$ to the whole ball $B=\{y:\, |y|<\delta\}$, in order to have
the equation above satisfied in $B$. More precisely, for $y_n>0$ we define
\begin{gather*}
u(y',-y_n)= -u(y',y_n), \\
g_i'(u_0(y',-y_n))=-g_i'(u_0(y',y_n)).
\end{gather*}
Moreover, denoting by $e_{lkj}$ the product $\frac{\partial
y_l}{\partial x_j} \frac{\partial y_k}{\partial x_j}$, we also
extend $e_{lkj}$ as
$$
e_{lkj}(y',-y_n)= \begin{cases}
e_{lkj}(y',y_n),  & l,k<n \; l=k=n, \\
-e_{lkj}(y',y_n),  & \mbox{otherwise}.
\end{cases}
$$
With this extensions it is possible to verify that:
\begin{multline} \label{ecub}
 \int_{B} \Big(1+\Big|\sum_{i=1}^{m}\sum_{j,l,k=1}^{n}
e_{lkj}\frac{\partial u^i}{\partial y_l}\frac{\partial
u^i}{\partial y_k}\Big|^ {\frac{p-2}2}\Big)
\sum_{i=1}^{m}\sum_{j,l,k=1}^{n} e_{lkj}\frac{\partial
u^i}{\partial y_l}\frac{\partial \phi^i}{\partial y_k}  \,dy \\
+ \int_{B} \sum_{i=1}^{m} g_i'(u_0)\phi^i  \, dy =0,
\end{multline}
for any $\phi\in C_0^{1}(B,\mathbb{R}^m)$.
 Setting $b(\eta,\eta')=  \sum_{i,j=1}^{m}
\sum_{l,k=1}^{n}\gamma_{lk}\delta_{ij} \eta^i_l {\eta'}^j_k$,
where $\gamma_{lk}=\sum_{j=1}^{n} e_{lkj}$, and
$A(t)=1+|t|^{\frac{p-2}2}$, (\ref{ecub}) becomes
\[
 \int_{B} A\left(b(\nabla u,\nabla u)\right) b(\nabla u,\nabla
\phi) \ dy  + \int_{B} \sum_{i=1}^{m} g_i'(u_0)\phi^i \,dy=0, \quad
  \forall \phi\in C_0^{1}(B,\mathbb{R}^m)
\]
We now prove that $(\gamma_{lk})$ is positive definite.
Indeed, it is sufficient to observe that in $D$
\[
e_{lkj}=\begin{cases}
\delta_{lj}\delta_{kj}, & l,k<n,\\
-\delta_{lj}\frac{\partial a}{\partial x_j}, & l<n, k=n,\\
-\delta_{kj}\frac{\partial a}{\partial x_j},  & k<n, l=n,\\
(\frac{\partial a}{\partial x_j})^2,  & l=k=n,
\end{cases}
\]
in the case $j<n$, while $e_{lkn}=\delta_{ln}\delta_{kn}$. This
implies that
$$
\gamma_{lk}= \begin{cases}
1, & l=k<n,\\
-\frac{\partial a}{\partial x_l},  & l<n, k=n,\\
-\frac{\partial a}{\partial x_k},  & l=n, k<n,\\
1+|\nabla a|^2,  & l=k=n, \\
0,  & \mbox{otherwise}.
\end{cases}
$$
Consequently there exist two positive constants $\lambda,
\lambda'$ such that
\begin{gather*}
b(\eta,\eta)\geq \lambda |\eta|^2, \ \forall \eta\in\mathbb{R}^{nm},\\
\sum_{i,j=1}^{m}\delta_{ij} +
\sum_{l,k=1}^{n}\|\gamma_{lk}\|_{C^1(\overline B)} \leq \lambda'.
\end{gather*}
For every $t>0$,
\begin{gather*}
\lambda (1+t)^{p-2} \leq A(t^2) \leq \lambda' (1+t)^{p-2},\\
(\lambda-\frac12)A(t)\leq t\frac{\partial A}{\partial t}(t)\leq \lambda' A(t),\\
t^2\left|\frac{\partial^2 A}{\partial t^2}(t)\right|\leq \lambda' A(t),\\
\sum_{i=1}^m |g'_i(u_0(y))|\leq \lambda', \ \forall y\in B.\\
\end{gather*}
 In particular we can apply the interior estimate in
\cite{tolksdorfs} to get that $u\in C^1(B,\mathbb{R}^m)$ and so the proof
of the theorem.
\end{proof}

\section{Critical groups computations}

The functional $I$ is $C^2$ on $W^{1,p}_0(\Omega, \mathbb{R}^m)$ and
critical points of $I$ on $W^{1,p}_0(\Omega, \mathbb{R}^m)$ are weak
solutions to \eqref{P}. In fact, since
$$
I(u)=\frac1p\int_\Omega \Big(\sum_{i=1}^{m}\sum_{\alpha=1}^{n}
\big(\frac{\partial u^i}{\partial x_\alpha}\big)^2
\Big)^{\frac{p}2}\, dx
+ \frac12\int_\Omega \sum_{i=1}^{m}\sum_{\alpha=1}^{n}
\big(\frac{\partial u^i}{\partial x_\alpha}\big)^2 \,dx +
\int_\Omega g(u)\, dx,
$$
we easily have for any $u,v \in W_0^{1,p}(\Omega,\mathbb{R}^m)$
\begin{align*}
\langle I'(u),v\rangle  = & \int_\Omega \Big(\sum_{i=1}^{m}\sum_{\alpha=1}^{n}
\big(\frac{\partial u^i}{\partial x_\alpha}\big)^2
\Big)^{\frac{p-2}2}\sum_{i=1}^{m}\sum_{\alpha=1}^{n}
\frac{\partial u^i}{\partial x_\alpha}\frac{\partial v^i}{\partial x_\alpha} \,
dx \\
& + \int_\Omega \sum_{i=1}^{m}\sum_{\alpha=1}^{n}
\frac{\partial u^i}{\partial x_\alpha}\frac{\partial v^i}{\partial x_\alpha} \,dx
 + \int_\Omega \sum_{i=1}^{m} g'_i(u) v^i dx, .
\end{align*}
This will be expressed briefly as
$$
\langle I'(u),v\rangle= \int_\Omega \left(1+\left|\nabla u
\right|^{p-2}\right) \left(\nabla u | \nabla v \right)
dx+ \int_\Omega \left( g'(u) | v\right) \,dx.
$$
It is easy to prove that the second order differential of $I$
in $u$ is given by
\begin{align*}
     \langle I''(u)\, v,w \rangle  =&
     \int_\Omega (1+ |\nabla u|^{p-2}) (\nabla v |\nabla w)\, dx \\
     %\label{Silvia}
& + (p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u|\nabla v)
(\nabla u| \nabla w)\, dx  +  \int_\Omega (g''(u)v|w)\, dx
\end{align*}
for any $v,w \in W_0^{1,p}(\Omega, \mathbb{R}^m)$.


 Now let us fix $u \in W^{1,p}_0(\Omega, \mathbb{R}^m)$ a
critical point of $I$. As mentioned in the introduction, the
operator $I''(u)$ is not a Fredholm operator, thus any generalized
splitting Morse lemma fails. To compute the critical groups in the
critical point $u$, we introduce an auxiliary Hilbert space,
depending on the critical point $u$, in which
$W_0^{1,p}(\Omega,\mathbb{R}^m)$ can be embedded, so that a natural
splitting can be obtained.
 By Theorem~\ref{bin}, we have that $u \in
C^{1}(\overline \Omega,\mathbb{R}^m)$. Therefore the vector function
$r^i(x)= |\nabla u(x)|^{(p-4)/2} \nabla u^i(x)$ belongs to
$C^0(\overline \Omega,\mathbb{R}^m)$, for each $i=1,\dots,m$. We write
$r=(r^1,\dots,r^m)$. Let $H_r$ be the closure of $C^{\infty}_0
(\Omega,\mathbb{R}^m)$ under the scalar product
\[
(v,w)_r = \int_\Omega (1+ |r|^2) (\nabla v |\nabla w) \, \,dx + (p-2)  (r |\nabla v) (r | \nabla w) \, dx
\]
and let $\langle \cdot , \cdot
\rangle_r : H_r^* \times H_r \to \mathbb{R}$ denote the duality pairing in
$H_r$.


 We emphasize that the space $H_r$ is $W_0^{1,2}(\Omega,\mathbb{R}^m)$
equipped by an equivalent Hilbert structure, which depends on the
critical point $u$, being suggested by $I''(u)$ itself. In such a
way $W_0^{1,p}(\Omega,\mathbb{R}^m) \subset H_r$ continuously.
 Furthermore $I''(u)$ can be extended to a Fredholm
operator
     $L_r : H_r \to  H_r^*$ defined by
setting
\[
\langle L_r v,w \rangle_r =  (v,w)_r \ + \langle K v,w \rangle_r
\]
where $ \langle Kv,w \rangle_r = \int_\Omega (g''(u)v| w)\,dx$
for any $v,w \in H_r$. As $L_r$ is a compact
perturbation of the Riesz isomorphism from $H_r$ to $H^*_r$, then
$L_r$ is a Fredholm operator with index zero. We can consider the
natural splitting \[ H_r= H^- \oplus H^0 \oplus H^+ \] where
$H^-, H^0, H^+$ are, respectively, the negative, null, and
positive spaces, according to the spectral decomposition of $L_r$
in $L^2(\Omega,\mathbb{R}^m)$.


 Furthermore, denoting by $\| \cdot \|_r$ the norm
induced by $(\cdot, \cdot)_r$, it is obvious that there exists $c> 0$ such that
\[
\langle L_r v,v \rangle_r + c \int_{\Omega} |v|^2 \, dx
\geq \|v\|_r^2 \quad \forall v \in H_r.
\]
 Hence it follows that there exists $\mu >0$ such that
\begin{equation}\label{dis}
 {\langle L_r v,v \rangle}_r \,
\geq \mu \|v\|_r^2 \quad \forall v \in H^+.
\end{equation}
 We claim now that
$H^- \oplus H^0 \subset W_0^{1,p}(\Omega,\mathbb{R}^m)$.
Indeed, this is a direct consequence of the fact that every $v \in
H_r$ which is a weak solution of the equation $L_r v + \eta v=0$,
belongs to $W_0^{1,p}(\Omega,\mathbb{R}^m)$. For such a solution $v$,we have
\begin{align*}
\int_\Omega (1+ |r|^2) (\nabla v |\nabla w) \, dx +
(p-2)  (r |\nabla v) (r | \nabla w) \, dx&\\
+ \int_\Omega (g''(u)v| w) \, dx + \int_\Omega (\eta v|
w) \, dx &=0,
\end{align*}
or equivalently
\begin{align*}
\int_\Omega (1+ |r|^2) \sum_{i=1}^{m}\sum_{\alpha=1}^{n}
\frac{\partial v^i}{\partial x_\alpha}\frac{\partial w^i}{\partial
x_\alpha} dx&\\
+ (p-2)\int_\Omega
\Big(\sum_{i=1}^{m}\sum_{\alpha=1}^{n} r_\alpha^i \frac{\partial
v^i}{\partial x_\alpha}\Big)
\Big(\sum_{j=1}^{m}\sum_{\beta=1}^{n} r_\beta^j\frac{\partial
w^j}{\partial x_\beta}\Big) \, dx&
\\
+ \int_\Omega \sum_{i=1}^{m}\sum_{j=1}^{m} D_{ij}g(u)v^j w^i \,
dx + \int_\Omega \sum_{i=1}^{m} \eta v^i w^i \,
dx&=0.
\end{align*}
After re-ordering the sums, we have that for all $w\in
W^{1,2}_0(\Omega,\mathbb{R}^m)$,
\begin{equation} \label{quit}
\begin{aligned}
\int_\Omega \sum_{i,j=1}^m\sum_{\alpha,\beta=1}^n \Bigl(
\bigl(\delta_{ij}\delta_{\alpha\beta} (1+
|r|^2) + (p-2) r_\alpha^i r_\beta^j \bigr) D_\alpha v^i D_\beta w^j \Bigr)&
\\
+ \int_\Omega \sum_{i=1}^m \Big( \sum_{j=1}^m (D_{ij}g(u) + \eta \delta_{ij}) v^j
\Big)w^i &=0,
\end{aligned}
\end{equation}
Let us set $A_{\alpha\beta}^ {ij}(x) = \bigl(\delta_{ij}\delta_{\alpha\beta} (1+
|r|^2) + (p-2) r_\alpha^i r_\beta^j \bigr)$
and $l_i(x)= \sum_{j=1}^m (D_{ij}g(u) + \eta \delta_{ij}) v^j$, then $(\ref{quit})$ is
equivalent to
\[
\int_\Omega \sum_{i,j=1}^m\sum_{\alpha,\beta=1}^n A_{\alpha\beta}^
{ij}(x)D_\alpha v^i D_\beta w^j + \int_\Omega \sum_{i=1}^m l_i(x) w^i =0, \quad
 \forall w\in W^{1,2}_0(\Omega,\mathbb{R}^m).
\]
By Theorem \ref{bin} we have $A_{\alpha\beta}^ {ij}\in C^{0,\mu}
(\overline \Omega, \mathbb{R}^m)$ and clearly satisfies a strict
ellipticity condition, in fact
\[
\sum_{i,j}\sum_{\alpha,\beta} A_{\alpha\beta}^ {ij}(x)
\xi_\alpha^i \xi_\beta^j \geq |\xi|^2, \quad \forall \ x \in
\Omega.
\]
Moreover, for $v$ fixed in the definition of $l_i(x)$ we have that
$l_i\in L^q(\Omega)$ if $v\in L^q(\Omega,\mathbb{R}^m)$ for some $q>1$. So we can use
the result in \cite[pp. 73-74]{G} with $q=2$ to conclude that $v
\in W^{1,2^*}(\Omega,\mathbb{R}^m)$. Then we can choose $q=2^*$ and apply
again the result in an iterative scheme,  thus after a finite
number of steps we have $v \in W^{1,q}(\Omega,\mathbb{R}^m)$ for some $q>n$,
which implies in particular that $v$ is locally H\"older
continuous.

 At this point we can use \cite[Theorem~3.5]{G} to get
that $v\in C^{1,\mu}(\Omega,\mathbb{R}^m)$ and as a consequence $v \in
W_0^{1,p}(\Omega,\mathbb{R}^m)$.
Consequently, denoted by $W = H^+ \cap
W^{1,p}_0(\Omega, \mathbb{R}^m)$ and $V=H^- \oplus H^0$, we get the splitting
\begin{equation*} %\label{spli}
W^{1,p}_0(\Omega, \mathbb{R}^m)= V \oplus W
\end{equation*}
and, by (\ref{dis}) we infer
\begin{equation*}  %\label{C1}
     \langle I''(u)v,v \rangle \geq \mu \|v
\|^2_{r} \quad \forall v \in W.
\end{equation*}
In particular $m^*(I,u) =\mathop{\rm dim}V$ is finite.

 Following the arguments in Lemma 4.4 in \cite{CV}, it is
possible to prove a sort of local weak convexity along the
direction of $W$.  More precisely, for any $M>0$ there exist
$r_0>0$ and $C>0$ such that for any $z \in W^{1,p}_0(\Omega, \mathbb{R}^m)
\cap L^{\infty}(\Omega, \mathbb{R}^m)$ with $\|z\|_{\infty} \leq M$, $\|z
-u \| < r_0$, we have
\begin{equation}\label{wea}
\langle I''(z) w,w \rangle \geq C \|w\|_r^2 \quad \forall w \in W.
\end{equation}
An essential tool in these arguments is an abstract result, due
to Ioffe \cite{Ioffe}, concerning sequentially lower semicontinuity
of integral functionals with respect to mixed strong-weak convergence,
both in the scalar and vectorial case.
 The inequality $(\ref{wea})$ is sufficient to obtain a
finite-dimensional reduction.

\begin{Lemma}\label{rid}
There exist $r>0$ and $\rho \in ]0, r[$  such
that for any $v \in V \cap \overline B_\rho (0)$ there
exists one and only one
$\overline w \in W \cap \overline B_r (0)$ such that for any
$z \in W \cap \overline B_r (0)$ we have
\[
I(v + \overline w + u) \leq I(v + z + u).
\]
Moreover $\overline w$ is the only element in $W \cap \overline
B_r (0)$ such that $\langle I'(u + v + \overline w ), z \rangle
=0$ for all $z \in W$.
\end{Lemma}

\begin{proof}
Firstly, we observe that $0$ is a local minimum for $I$ along the
direction of $W$. This can be proved arguing as in \cite[Lemma 4.5]{CV}.
As in  Lemma \ref{linfty}, it is possible to prove an
uniform $L^{\infty}$-bound for the critical points of $I$ along
$W$, which are sufficiently close to $u$. The claim of
Lemma \ref{rid} can be finally deduced by similar arguments to
those used in \cite[Lemma 4.6]{CV}.
\end{proof}

\noindent {\em Proof of Theorem 1.2 and Theorem 1.3 completed.} We
can introduce the map
\begin{equation*}  %\label{psi}
\psi :  V \cap \overline B_\rho(0) \to W \cap B_{r}(0)
\end{equation*}
where $\psi(v)=\overline w$ is the unique minimum point of the
function $w \in W \cap \overline B_r(0) \mapsto I(u+v+w)$, and the
function
\begin{equation*}  %\label{phi}
\phi : V \cap \overline B_\rho(0) \to \mathbb{R}
\end{equation*}
defined by $\phi(v)=I(u+v+ \psi(v))$.
It is not difficult to show that the maps $\psi$ and $\phi$ are continuous.
 Using a suitable pseudogradient flow, like in section 5
of \cite{CV} it can be proved that
\begin{equation}\label{gruppi}
C_j(I,u)=C_j(\phi,0).
\end{equation}
In particular, if $I''(u)$ is injective, it can be deduced that
$0$ is a local maximum of $\phi$ in $V \cap \overline B_\rho(0)$,
so that by $(\ref{gruppi})$ Theorem~\ref{generalizzato} comes.

 More generally, not requiring the injectivity of
$I''(u)$, it is clear that $C_j(\phi,0)=\{0\}$ when $j\geq
m^*(I,u)+1= \mbox{\rm dim }V+1$. Finally \cite[Theorem 2.6]{lancelotti}
assures that $C_j(\phi,0)=\{0\}$ if $j\leq m(I,u)-1$ and thus
Theorem \ref{indicelargo} follows.

\subsection*{Acknowledgement} This work has been developed
while the first author was visiting the Dipartimento
Interuniversitario di Matematica, Universit\`{a} degli Studi di
Bari-Politecnico di Bari. He gratefully acknowledges the whole
department for the warm hospitality and the friendly atmosphere.
This work was supported by Azione Integrata Italia-Spagna
HI2000-0108.

\begin{thebibliography}{00}
%\bibitem{BFP}
%V. Benci, D. Fortunato, L. Pisani;
%{\em Solitons like solutions for a Lorentz invariant
%equations in dimension 3},
% Reviews Math. Phys., {\bf 3} (1998), 315--344.
%
%\bibitem{BDFP}
%V. Benci, P. D'Avenia, D. Fortunato, L. Pisani; {\em Solitons in
%several space dimensions: a Derrick's problem and infinitely many
%solutions},  Arch. Rat. Mech. Anal., {\bf 154} (2000),
%297--324.

\bibitem{2}
K.C. Chang, {\em Morse Theory on Banach space and its applications
to partial differential equations},   Chin. Ann. of Math., {\bf
4B} (1983), 381--399.

\bibitem{3}
K.C. Chang, {\em Morse theory in nonlinear analysis}, in Nonlinear
Functional Analysis and Applications to Differential Equations, A.
Ambrosetti, K. C. Chang, I. Ekeland Eds., Word Scientific
Singapore, 1998.

\bibitem{CV1}
S. Cingolani, G. Vannella, {\em Some results on critical groups
estimates for a class of functionals defined on Sobolev Banach
spaces},  Rend. Mat. Acc. Naz. Lincei, {\bf 12}, 199--201
(2001).

\bibitem{CV}
S. Cingolani, G. Vannella, {\em Critical groups computations on a
class of Sobolev Banach spaces via Morse index}, Ann. Inst. H.
Poincar\'{e}: analyse non lin\'{e}aire, {\bf 2}, 271--292 (2003).

\bibitem{CV2}
S. Cingolani, G. Vannella, {\em Marino-Prodi perturbation type
results and Morse indices of minimax critical points for a class
of functionals defined in Banach spaces},  preprint.

\bibitem{egnell} H. Egnell,
{\em Existence and Nonexistence Results for $m$-Laplace
Equations Involving Critical Sobolev Exponents},
Arch. Rational Mech. Anal., {\bf 104} (1988), 57-77.

\bibitem{G}
M. Giaquinta, {\em Introduction to Regularity Theory for Nonlinear
Elliptic Systems},  Lectures in Mathematics. Birkhauser, 1993.

\bibitem{HS}
E. Hewitt, K. Stromberg, {\em Real and Abstract Analysis},
Springer-Verlag, New York Heidelberg Berlin, 1975.

\bibitem{Ioffe}
A. D. Ioffe, {\em On lower semicontinuity of integral functionals I and
II}, S.I.A.M. J. Control Optim., {\bf 15} (1977), 521--538,
and 991--1000.

%\bibitem{lu}
%O. A. Ladyzhenskaya, N. N. Ural'tseva, {\em Linear and Quasilinear
%Elliptic Equations}, Academic Press, New York London, 1968.

\bibitem{lancelotti}
S. Lancelotti, {\em Morse index estimates for continuous functionals
associated with quasilinear elliptic equations},
 Advances Diff. Eqs., {\bf 7} (2002), 99--128.

\bibitem{MP}
A. Marino, G. Prodi, {\em Metodi perturbativi nella teoria di Morse},
Boll. U.M.I., (4) {\bf 11} Suppl. fase 3 (1975), 1--32.


\bibitem{stam}
G. Stampacchia, {\em \'Equations Elliptiques du second ordre \'a Coefficients
Discontinus}, Les Presses de L'Universit\'e de Montr\'eal., 1965.

\bibitem{tolksdorfs}
 P. Tolksdorf, {\em Everywhere-Regularity for some quasilinear
systems with a lack of ellipticity},  Annali di Matematica Pura
ed Applicata, {\bf 134} (1983), 241-266.

\bibitem{tolksdorf}
P. Tolksdorf, {\em On the Dirichlet problem for Quasilinear
Equations in Domains with Conical Boundary Points},
Communications in Partial Differential Equations, {\bf 8} (1983),
773-811.

\bibitem{9}
A. J. Tromba, {\em A general approach to Morse theory},  J. Diff.
Geometry, {\bf 12} (1977), 47--85.

\bibitem{10}
K. Uhlenbeck, {\em Morse theory on Banach manifolds},  J. Funct.
Anal., {\bf 10} (1972), 430--445.

\end{thebibliography}

\end{document}
)