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\markboth{\hfil Resonance problems  \hfil EJDE--2002/36}
{EJDE--2002/36\hfil Kanishka Perera \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 36, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Resonance problems with respect to the Fu\v c\'\i k
  spectrum of the $p$-Laplacian
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 35P99, 47J30, 58E50.
 \hfil\break\indent 
{\em Key words:} $p$-Laplacian problems, Fu\v c\'\i k spectrum, resonance,
variational methods, \hfil\break\indent linking. \hfil\break\indent 
\copyright 2002 Southwest Texas State University. \hfil\break\indent 
Submitted January 9, 2002. Published April 23, 2002.} }
\date{}
%
\author{Kanishka Perera}
\maketitle

\begin{abstract}
We solve resonance problems with respect to the Fu\v c\'\i k
spectrum of the $p$-Laplacian using variational methods.
\end{abstract}

\newcommand{\bigip}[2]{\big(#1,#2\big)}
\newcommand{\norm}[2]{\|#1\|_{#2}}
\newcommand{\seq}[1]{\left(#1\right)}
\newcommand{\set}[1]{\left\{#1\right\}}
\newenvironment{enumeratepr}{\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{(\roman{enumi}).}}{\end{enumerate}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\numberwithin{equation}{section}

\section{Introduction} \label{S1}

Consider the quasilinear elliptic boundary value problem
\begin{equation} \label{101}
- \Delta_p\, u = a\, (u^+)^{p-1} - b\, (u^-)^{p-1} + f(x,u),
\quad u \in W^{1,\, p}_0(\Omega)
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb {R}^n,\, n \ge 1$,
$\Delta_p\, u = \mathop{\rm div} \big(|\nabla u|^{p-2}\, \nabla u\big)$ is
the $p$-Laplacian, $1 < p < \infty$, $u^\pm = \max \set{\pm\, u,
0}$, and $f$ is a Carath\'{e}odory function on $\Omega \times \mathbb {R}$
satisfying a growth condition
\begin{equation} \label{102}
|f(x,t)| \le V(x)^{p-q}\, |t|^{q-1} + W(x)^{p-1}
\end{equation}
with $1 \le q < p$ and $V,\, W \in L^p(\Omega)$. The set
$\Sigma_p$ of those points $(a,b) \in \mathbb {R}^2$ for which the
asymptotic problem
\begin{equation} \label{103}
- \Delta_p\, u = a\, (u^+)^{p-1} - b\, (u^-)^{p-1}, \quad u \in
W^{1,\, p}_0(\Omega)
\end{equation}
has a nontrivial solution is called the Fu\v c\'\i k  spectrum of
the $p$-Laplacian on $\Omega$. The nonresonance case for problem
\eqref{101}, $(a,b) \notin \Sigma_p$, was recently studied by
Cuesta, de Figueiredo, and Gossez \cite{CudeFiGo} and the author
\cite{Pe6, Pe5}. The symmetric resonance case, $a = b \in \sigma(-
\Delta_p)$, was considered by Dr{\'a}bek and Robinson \cite{DrRo}.
The purpose of the present paper is to study the general resonance
case $(a,b) \in \Sigma_p$.

The Fu\v c\'\i k  spectrum was introduced in the semilinear
case,  $p = 2$, by Dancer \cite{Da3} and Fu\v c\'\i k  \cite{Fu1}
who recognized its significance for the solvability of problems
with jumping nonlinearities. In the semilinear ODE case $p = 2,\,
n = 1$, Fu\v c\'\i k  \cite{Fu1} showed that $\Sigma_2$ consists
of a sequence of hyperbolic like curves passing through the points
$(\lambda_l,\lambda_l)$, where $\set{\lambda_l}_{l \in
\mathbb{N}}$ are the eigenvalues of $- \Delta$, with one or two
curves going through each point. Dr{\'a}bek \cite{Dr} has
recently shown that $\Sigma_p$ has this same general shape for
all $p > 1$ in the ODE case.

In the PDE case, $n \ge 2$, much of the work to date on $\Sigma_p$
has been for the semilinear case. It is now known that $\Sigma_2$
consists, at least locally, of curves emanating from the points
$(\lambda_l,\lambda_l)$ (see, e.g., \cite{Ca, CuGo, Da3, deFiGo,
Fu1, GaKa, La, LaMc1, LaMc2, Ma, MaMa}). Schechter \cite{Sc7} has
shown that $\Sigma_2$ contains two continuous and strictly
decreasing curves through $(\lambda_l,\lambda_l)$, which may
coincide, such that the points in the square
$(\lambda_{l-1},\lambda_{l+1})^2$ that are either below the lower
curve or above the upper curve are not in $\Sigma_2$, while the
points between them may or may not belong to $\Sigma_2$ when they
do not coincide.

In the quasilinear PDE case, $p \ne 2,\, n \ge 2$, it is known
that the first eigenvalue $\lambda_1$ of $- \Delta_p$ is positive,
simple, and admits a positive eigenfunction $\varphi_1$ (see
Lindqvist \cite{Lin1}). Hence $\Sigma_p$ contains the two lines
$\lambda_1 \times \mathbb {R}$ and $\mathbb {R} \times
\lambda_1$. In addition, $\sigma(- \Delta_p)$ has an unbounded
sequence of variational eigenvalues $\set{\lambda_l}$ satisfying
a standard min-max characterization, and $\Sigma_p$ contains the
corresponding sequence of points $\set{(\lambda_l,\lambda_l)}$. A
first nontrivial curve in $\Sigma_p$ through
$(\lambda_2,\lambda_2)$ asymptotic to $\lambda_1 \times \mathbb
{R}$ and $\mathbb {R} \times \lambda_1$ at infinity was recently
constructed and variationally characterized by a mountain-pass
procedure by Cuesta, de Figueiredo, and Gossez \cite{CudeFiGo}.
More recently, unbounded sequences of curves $\set{C^\pm_l}$ in
$\Sigma_p$ (analogous to the lower and upper curves of Schechter)
have been constructed and variationally characterized by min-max
procedures by Micheletti and Pistoia \cite{MiPi} for $p \ge 2$
and by the author \cite{Pe5} for all $p
> 1$.

Let us also mention that some Morse theoretical aspects of the
Fu\v c\'\i k  spectrum have been studied in Dancer \cite{Da2},
Dancer and Perera \cite{DaPe}, Perera and Schechter \cite{PeSc8,
PeSc2, PeSc3, PeSc6, PeSc5, PeSc9}, and Li, Perera, and Su
\cite{LiPeSu2}.

Denote by $N$ the set of nontrivial solutions of \eqref{103}, and
set
\begin{equation} \label{104}
F(x,t) := \int_0^t f(x,s)\, ds, \quad H(x,t) := p\, F(x,t) - t
f(x,t).
\end{equation}
The main result of this paper is:

\begin{theorem} \label{T101}
The problem \eqref{101} has a solution if
\begin{enumeratepr}
\item \label{H101} $(a,b) \in C^+_l$ and $\int_\Omega
H(x,u_j) \to + \infty$, or
\item \label{H102} $(a,b) \in C^-_l$ and $\int_\Omega
H(x,u_j) \to - \infty$
\end{enumeratepr}
for every sequence $\seq{u_j}$ in $W^{1,\, p}_0(\Omega)$ such that
$\norm{u_j}{} \to \infty$ and $u_j/\norm{u_j}{}$ converges to some
element of $N$.
\end{theorem}

As is usually the case in resonance problems, the main difficulty
here is the lack of compactness of the associated variational
functional. We will overcome this difficulty by constructing a
sequence of approximating nonresonance problems, finding
approximate solutions for them using linking and min-max type
arguments, and passing to the limit. But first we give some
corollaries. In what follows, $\seq{u_j}$ is as in the theorem,
i.e., $\rho_j := \norm{u_j}{} \to \infty$ and $v_j := u_j/\rho_j
\to v \in N$.

First we give simple pointwise assumptions on $H$ that imply the
limits in the theorem.

\begin{corollary} \label{T102}
Problem \eqref{101} has a solution in the following cases:
\begin{enumeratepr}
\item \label{H103} $(a,b) \in C^+_l$, $H(x,t) \to + \infty$ a.e. as
$|t| \to \infty$, and $H(x,t) \ge - C(x)$,
\item \label{H104} $(a,b) \in C^-_l$, $H(x,t) \to - \infty$ a.e. as
$|t| \to \infty$, and $H(x,t) \le C(x)$
\end{enumeratepr}
for some $C \in L^1(\Omega)$.
\end{corollary}

Note that this corollary makes no reference to $N$.

\begin{proof}
If \eqref{H103} holds, then $H(x,u_j(x)) = H(x,\rho_j\, v_j(x))
\to + \infty$ for a.e. $x$ such that $v(x) \ne 0$ and
$H(x,u_j(x)) \ge - C(x)$, so
\begin{equation} \label{105}
\int_\Omega H(x,u_j) \ge \int_{v \ne 0} H(x,u_j) - \int_{v = 0}
C(x) \to + \infty
\end{equation}
by Fatou's lemma. Similarly, $ \int_\Omega H(x,u_j)
\to - \infty$ if \eqref{H104} holds.
\end{proof}

Note that the above argument goes through as long as the limits in
\eqref{H103} and \eqref{H104} hold on subsets of $\set{x \in
\Omega : v(x) \ne 0}$ with positive measure. Now, taking $w =
v^+$ in
\begin{equation} \label{106}
\int_\Omega |\nabla v|^{p-2}\, \nabla v \cdot \nabla w =
\int_\Omega \Big[a\, (v^+)^{p-1} - b\, (v^-)^{p-1}\Big] w
\end{equation}
gives
\begin{align} \label{107}
\norm{v^+}{}^p & = \int_{\Omega_+} a\, (v^+)^p \le a\,
\norm{v^+}{p^\ast}^p\, \mu(\Omega_+)^{p/n} \notag\\
& \le a\, S^{-1}\, \norm{v^+}{}^p\, \mu(\Omega_+)^{p/n}
\end{align}
where $\Omega_+ = \set{x \in \Omega : v(x) > 0}$, $p^\ast =
np/(n - p)$ is the critical Sobolev exponent, $S$ is the best
constant for the embedding $W^{1,\, p}_0(\Omega) \hookrightarrow
L^{p^\ast}(\Omega)$, and $\mu$ is the Lebesgue measure in $\mathbb {R}^n$,
so
\begin{equation} \label{108}
\mu(\Omega_+) \ge \left(\frac{S}{a}\right)^{n/p}.
\end{equation}
A similar argument shows that
\begin{equation} \label{109}
\mu(\Omega_-) \ge \left(\frac{S}{b}\right)^{n/p}
\end{equation}
where $\Omega_- = \set{x \in \Omega : v(x) < 0}$, and hence
\begin{equation} \label{110}
\mu\Big(\set{x \in \Omega : v(x) = 0}\Big) \le \mu(\Omega) -
S^{n/p}\, \Big(a^{-n/p} + b^{-n/p}\Big).
\end{equation}
Thus,

\begin{corollary} \label{T103}
Problem \eqref{101} has a solution in the following cases:
\begin{enumeratepr}
\item \label{H105} $(a,b) \in C^+_l$, $H(x,t) \to + \infty$ in
$\Omega'$ as $|t| \to \infty$, and $H(x,t) \ge - C(x)$,
\item \label{H106} $(a,b) \in C^-_l$, $H(x,t) \to - \infty$ in
$\Omega'$ as $|t| \to \infty$, and $H(x,t) \le C(x)$
\end{enumeratepr}
for some $\Omega' \subset \Omega$ with $\mu(\Omega') > \mu(\Omega)
- S^{n/p} \left(a^{-n/p} + b^{-n/p}\right)$ and $C \in
L^1(\Omega)$.
\end{corollary}

Next note that
\begin{equation} \label{111}
\begin{split}
\underline{H}_+(x)\, & (v^+(x))^q + \underline{H}_-(x)\,
(v^-(x))^q
\le \liminf\, \frac{H(x,u_j(x))}{\rho_j^q}\\
& \hspace{-0.20in} \le \limsup\, \frac{H(x,u_j(x))}{\rho_j^q} \le
\overline{H}_+(x)\, (v^+(x))^q + \overline{H}_-(x)\, (v^-(x))^q
\end{split}
\raisetag{24pt}
\end{equation}
where
\begin{equation} \label{112}
\underline{H}_\pm(x) = \liminf_{t \to \pm \infty}\,
\dfrac{H(x,t)}{|t|^q}, \quad \overline{H}_\pm(x) = \limsup_{t \to
\pm \infty}\, \dfrac{H(x,t)}{|t|^q}.
\end{equation}
Moreover,
\begin{equation} \label{113}
\frac{|H(x,u_j(x))|}{\rho_j^q} \le (p + q)\, V(x)^{p-q}\,
|v_j(x)|^q + \frac{(p + 1)\, W(x)^{p-1}\, |v_j(x)|}{\rho_j^{q-1}}
\end{equation}
by \eqref{102}, so it follows that
\begin{equation} \label{114}
\begin{split}
\int_\Omega \underline{H}_+ (v^+)^q + \underline{H}_- & (v^-)^q
\le
\liminf\, \frac{\textstyle \int_\Omega H(x,u_j)}{\rho_j^q}\\
& \hspace{-0.3in} \le \limsup\, \frac{\textstyle \int_\Omega
H(x,u_j)}{\rho_j^q} \le \int_\Omega \overline{H}_+ (v^+)^q +
\overline{H}_- (v^-)^q.
\end{split}
\raisetag{24pt}
\end{equation}
Thus we have

\begin{corollary} \label{T104}
Problem \eqref{101} has a solution in the following cases:
\begin{enumeratepr}
\item \label{H107} $(a,b) \in C^+_l$ and $\int_\Omega
\underline{H}_+ (v^+)^q + \underline{H}_- (v^-)^q > 0 \quad
\forall v \in N$,
\item \label{H108} $(a,b) \in C^-_l$ and $ \int_\Omega
\overline{H}_+ (v^+)^q + \overline{H}_- (v^-)^q < 0 \quad \forall
v \in N$.
\end{enumeratepr}
\end{corollary}

Finally we note that if
\begin{equation} \label{115}
\dfrac{t f(x,t)}{|t|^q} \to f_\pm(x) \quad \text{a.e. as } t \to \pm \infty,
\end{equation}
then
\begin{equation} \label{116}
\frac{F(x,t)}{|t|^q} = \frac{1}{|t|^q} \int_0^t \left[\frac{s
f(x,s)}{|s|^q} - f_\pm(x)\right] \! |s|^{q-2} s\, ds +
\frac{f_\pm(x)}{q} \to \frac{f_\pm(x)}{q}
\end{equation}
and hence
\begin{equation} \label{117}
\dfrac{H(x,t)}{|t|^q} \to \bigg(\dfrac{p}{q} - 1\bigg) f_\pm(x),
\end{equation}
so Corollary \ref{T104} implies

\begin{corollary} \label{T105}
Problem \eqref{101} has a solution in the following cases:
\begin{enumeratepr}
\item \label{f101} $(a,b) \in C^+_l$ and $ \int_\Omega
f_+ (v^+)^q + f_- (v^-)^q > 0 \quad \forall v \in N$,
\item \label{f102} $(a,b) \in C^-_l$ and $\int_\Omega
f_+ (v^+)^q + f_- (v^-)^q < 0 \quad \forall v \in N$.
\end{enumeratepr}
\end{corollary}

\section{Preliminaries on the Fu\v c\'\i k  Spectrum} \label{S2}

As in Cuesta, de Figueiredo, and Gossez \cite{CudeFiGo}, the
points in $\Sigma_p$ on the line parallel to the diagonal $a = b$
and passing through $(s,0)$ are of the form $(s + c_+,c_+)$ (resp.
$(c_-,s + c_-)$) with $c_\pm$ a critical value of
\begin{equation} \label{201}
J^\pm_s(u) = \int_\Omega |\nabla u|^p - s\, (u^\pm)^p, \quad u \in
S = \set{u \in W^{1,\, p}_0(\Omega) : \norm{u}{p} = 1}
\end{equation}
and $J^\pm_s$ satisfies the Palais-Smale compactness condition.
Since $\Sigma_p$ is clearly symmetric with respect to the
diagonal, we may assume that $s \ge 0$. In particular, the
eigenvalues of $- \Delta_p$ on $W^{1,\, p}_0(\Omega)$ correspond
to the critical values of the even functional $J = J^\pm_0$. As
observed in Dr{\'a}bek and Robinson \cite{DrRo}, we can define an
unbounded sequence of critical values of $J$ by
\begin{equation} \label{202}
\lambda_l := \inf_{A \in {\cal F}_l} \max_{u \in A} J(u), \quad l
\in \mathbb{N}
\end{equation}
where
\begin{equation} \label{203}
{\cal F}_l = \set{A \subset S : \text{there is a continuous odd
surjection } h : S^{l-1} \to A}
\end{equation}
and $S^{l-1}$ is the unit sphere in $\mathbb {R}^l$, although it is not
known whether this gives a complete list of eigenvalues.

Suppose that $l \ge 2$ is such that $\lambda_l > \lambda_{l-1}$
and let $0 < \varepsilon < \lambda_l - \lambda_{l-1}$ be given. By
\eqref{202}, there is an $A^{l-2} \in {\cal F}_{l-1}$ such that
\begin{equation} \label{204}
\max_{u \in A^{l-2}} J(u) < \lambda_{l-1} + \varepsilon.
\end{equation}
Let $h_{l-2} : S^{l-2} \to A^{l-2}$ be any continuous odd
surjection and let
\begin{align} \label{205}
{\cal F}_l^+ = \big\{&A_+ \subset S : \text{there is a continuous
surjection } h : S^{l-1}_+ \to A_+ \notag\\ & \text{such that }
h|_{S^{l-2}} = h_{l-2}\big\}
\end{align}
where $S^{l-1}_+$ is the upper hemisphere of $S^{l-1}$ with
boundary $S^{l-2}$. Then ${\cal F}_l^+$ is a homotopy-stable
family of compact subsets of $S$ with closed boundary $A^{l-2}$,
i.e,
\begin{enumeratepr}
\item every set $A_+ \in {\cal F}_l^+$ contains $A^{l-2}$,
\item for any set $A_+ \in {\cal F}_l^+$ and any $\eta \in
C([0,1] \times S; S)$ satisfying $\eta(t,u) = u$ for all $(t,u)
\in (\set{0} \times S) \cup ([0,1] \times A^{l-2})$ we have that
$\eta(\set{1} \times A) \in {\cal F}_l^+$.
\end{enumeratepr}

For $s \in I_l^\varepsilon := [0,\lambda_l - \lambda_{l-1} -
\varepsilon]$, set
\begin{equation} \label{206}
c^\pm_l(s) := \inf_{A_+ \in {\cal F}_l^+} \max_{u \in A_+}
J^\pm_s(u).
\end{equation}
If $c^\pm_l(s) < \lambda_l - s$, taking $A_+ \in {\cal F}_l^+$
with
\begin{equation} \label{207}
\max_{u \in A_+} J^\pm_s(u) < \lambda_l - s
\end{equation}
and setting $A = A_+ \cup (- A_+)$ we get a set in ${\cal F}_l$
for which
\begin{equation} \label{208}
\max_{u \in A} J(u) = \max_{u \in A_+} J(u) \le \max_{u \in A_+}
J^\pm_s(u) + s < \lambda_l,
\end{equation}
a contradiction. Thus
\begin{equation} \label{209}
c^\pm_l(s) \ge \lambda_l - s \ge \lambda_{l-1} + \varepsilon >
\max_{u \in A^{l-2}} J(u) \ge \max_{u \in A^{l-2}} J^\pm_s(u),
\end{equation}
and it follows from Theorem 3.2 of Ghoussoub \cite{Gh2} that
$c^\pm_l(s)$ is a critical value of $J^\pm_s$. Hence
\begin{equation} \label{210}
C^\pm_l := \set{(s + c^\pm_l(s),c^\pm_l(s)) : s \in
I_l^\varepsilon} \cup \{(c^\pm_l(s),s + c^\pm_l(s)) : s \in
I_l^\varepsilon\} \subset \Sigma_p.
\end{equation}
Note that \eqref{209} implies $c^\pm_l(0) \ge \lambda_l \to
\infty$.

\section{Proof of Theorem \ref{T101}} \label{S3}

As is well-known, solutions of \eqref{102} are the critical
points of
\begin{equation} \label{301}
\Phi(u) = \int_\Omega |\nabla u|^p - a\, (u^+)^p - b\, (u^-)^p -
p\, F(x,u), \quad u \in W^{1,\, p}_0(\Omega).
\end{equation}
We only consider \eqref{H101} as the proof for \eqref{H102} is
similar. Let $(a,b) = (s + c^+_l(s),c^+_l(s))$, $s \ge 0$ and
\begin{equation} \label{302}
\Phi_j(u) = \Phi(u) + \frac{1}{j}\, \int_\Omega |u|^p =
\int_\Omega |\nabla u|^p - s\, (u^+)^p - \left(c^+_l(s) -
\frac{1}{j}\right) |u|^p - p\, F(x,u).
\end{equation}
First we show that, for sufficiently large $j$, there is a $u_j
\in W^{1,\, p}_0(\Omega)$ such that
\begin{equation} \label{303}
\norm{u_j}{} \,\norm{\Phi_j'(u_j)}{} \to 0, \quad
\inf \Phi_j(u_j) > - \infty.
\end{equation}

Let $\varepsilon$ and $A^{l-2}$ be as in Section \ref{S2}. By
\eqref{209},
\begin{equation} \label{304}
\max_{u \in A^{l-2}} J^\pm_s(u) \le c^\pm_l(s) - \frac{2}{j}
\end{equation}
for sufficiently large $j$. For such $j$, $u \in A^{l-2}$, and $R
> 0$,
\begin{align} \label{305}
\Phi_j(Ru) & = R^p \left[J^+_s(u) - \left(c^+_l(s) -
\frac{1}{j}\right)\right] - \int_\Omega p\, F(x,Ru)
\notag\\
& \le - \frac{R^p}{j} + p\, \Big(\norm{V}{p}^{p-q}\, R^q +
\norm{W}{p}^{p-1}\, R\Big)
\end{align}
by \eqref{102}, so
\begin{equation} \label{306}
\max_{u \in A^{l-2}} \Phi_j(Ru) \to - \infty \quad \text{as } R \to \infty.
\end{equation}
Next let
\begin{equation} \label{307}
F = \set{u \in W^{1,\, p}_0(\Omega) : J^+_s(u) \ge c^+_l(s)\,
\norm{u}{p}^p}.
\end{equation}
For $u \in F$,
\begin{equation} \label{308}
\Phi_j(u) \ge \frac{\norm{u}{p}^p}{j} - p\,
\Big(\norm{V}{p}^{p-q}\, \norm{u}{p}^q + \norm{W}{p}^{p-1}\,
\norm{u}{p}\Big),
\end{equation}
so
\begin{equation} \label{309}
\inf_{u \in F} \Phi_j(u) \ge C := \min_{r \ge 0}
\bigg[\frac{r^p}{j} - p\, \Big(\norm{V}{p}^{p-q}\, r^q +
\norm{W}{p}^{p-1}\, r\Big)\bigg] > - \infty.
\end{equation}
Now use \eqref{306} to fix $R > 0$ so large that
\begin{equation} \label{310}
\max \Phi_j(B) < C
\end{equation}
where $B = \set{Ru : u \in A^{l-2}}$.

Next consider the homotopy-stable family of compact subsets of $X$
with boundary $B$ given by
\begin{align} \label{410}
{\cal F} = \big\{&A \subset X : \text{there is a continuous
surjection } h : S^{l-1}_+ \to A \notag\\
& \text{such that } h|_{S^{l-2}} = R\, h_{l-2}\big\}
\end{align}
where $h_{l-2}$ is as in Section \ref{S2}. We claim that the set
$F$ is dual to the class $\cal F$, i.e.,
\begin{equation} \label{411}
F \cap B = \emptyset, \quad F \cap A \ne \emptyset \quad \forall
A \in {\cal F}.
\end{equation}
It is clear from \eqref{309} and \eqref{310} that $F \cap B =
\emptyset$. Let $A \in {\cal F}$. If $0 \in A$, then we are done.
Otherwise, denoting by $\pi$ the radial projection onto $S$,
$\pi(A) \in {\cal F}_l^+$ and hence
\begin{equation} \label{412}
\max_{u \in \pi(A)} J^+_s(u) \ge c^+_l(s),
\end{equation}
so $F \cap \pi(A) \ne \emptyset$. But this implies $F \cap A \ne
\emptyset$.

Now it follows from a deformation argument of Cerami \cite{Ce}
that there is a $u_j$ such that
\begin{equation} \label{314}
\norm{u_j}{} \, \norm{\Phi_j'(u_j)}{} \to 0, \quad
|\Phi_j(u_j) - c_j| \to 0
\end{equation}
where
\begin{equation} \label{315}
c_j := \inf_{A \in {\cal F}} \max_{u \in A} \Phi_j(u) \ge C,
\end{equation}
from which \eqref{303} follows.

We complete the proof by showing that a subsequence of $\seq{u_j}$
converges to a solution of \eqref{101}. It is easy to see that
this is the case if $\seq{u_j}$ is bounded, so suppose that
$\rho_j := \norm{u_j}{} \to \infty$. Setting $v_j := u_j/\rho_j$
and passing to a subsequence, we may assume that $v_j \to v$
weakly in $W^{1,\, p}_0(\Omega)$, strongly in $L^p(\Omega)$, and
a.e. in $\Omega$. Then
\begin{align} \label{316}
\int_\Omega |\nabla v_j|^{p-2}\, \nabla v_j \cdot \nabla (v_j & -
v) = \frac{\bigip{\Phi_j'(u_j)}{v_j - v}}{p\, \rho_j^{p-1}}
+ \int_\Omega \bigg[\!\left(a - \frac{1}{j}\right)\!(v_j^+)^{p-1}
\notag\\
& - \left(b - \frac{1}{j}\right)\!(v_j^-)^{p-1} +
\frac{f(x,u_j)}{\rho_j^{p-1}}\, \bigg] (v_j - v) \to 0,
\end{align}
and we deduce that $v_j \to v$ strongly in $W^{1,\, p}_0(\Omega)$
(see, e.g., Browder \cite{Br}). In particular, $\norm{v}{} = 1$,
so $v \ne 0$. Moreover, for each $w \in W^{1,\, p}_0(\Omega)$,
passing to the limit in
\begin{align} \label{317}
\frac{\bigip{\Phi_j'(u_j)}{w}}{p\, \rho_j^{p-1}} = \int_\Omega
|\nabla v_j|^{p-2}\, & \nabla v_j \cdot \nabla w - \bigg[\!\left(a
- \frac{1}{j}\right)\!(v_j^+)^{p-1} \notag\\
& - \left(b - \frac{1}{j}\right)\!(v_j^-)^{p-1} +
\frac{f(x,u_j)}{\rho_j^{p-1}}\, \bigg] w \raisetag{24pt}
\end{align}
gives
\begin{equation} \label{318}
\int_\Omega |\nabla v|^{p-2}\, \nabla v \cdot \nabla w - \Big[a\,
(v^+)^{p-1} - b\, (v^-)^{p-1}\Big] w = 0,
\end{equation}
so $v \in N$. Thus,
\begin{equation} \label{319}
\frac{\bigip{\Phi_j'(u_j)}{u_j}}{p} - \Phi_j(u_j) = \int_\Omega
H(x,u_j) \to + \infty,
\end{equation}
contradicting \eqref{303}.

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\noindent\textsc{Kanishka Perera}\\
 Department of Mathematical Sciences\\
 Florida Institute of Technology\\
 Melbourne, FL 32901-6975, USA\\
e-mail: kperera@winnie.fit.edu\\
http://winnie.fit.edu/$\sim$kperera/
\end{document}