\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Minimax principles for critical-point theory \hfil EJDE--2000/18} {EJDE--2000/18\hfil A. R. El Amrouss \& M. Moussaoui \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~18, pp.~1--9. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Minimax principles for critical-point theory in applications to quasilinear boundary-value problems \thanks{ {\em Mathematics Subject Classifications:} 49J35, 35J65, 35B34. \hfil\break\indent {\em Key words and phrases:} Minimax methods, p-Laplacian, resonance. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted September 9, 1999. Published March 8, 2000.} } \date{} % \author{A. R. El Amrouss \& M. Moussaoui} \maketitle \begin{abstract} Using the variational method developed in \cite{E}, we establish the existence of solutions to the equation $-\Delta_p u = f(x,u)$ with Dirichlet boundary conditions. Here $\Delta_p $ denotes the p-Laplacian and $\int_0^s f(x,t)\,dt$ is assumed to lie between the first two eigenvalues of the p-Laplacian. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lem}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \section{Introduction} Consider the Dirichlet problem for the p-Laplacian $(p > 1)$, \begin{eqnarray} &-\Delta_p u = f(x,u) \quad\mbox{in }\Omega & \label{P}\\ &u = 0 \quad\mbox{on }\partial\Omega\,,& \nonumber \end{eqnarray} where $\Omega$ is a bounded domain in ${\mathbb R}^N$ with smooth boundary $\partial\Omega$. We assume that $f: \Omega \times {\mathbb R} \to {\mathbb R} $ is a Carath\'eodory function with subcritical growth; that is, $$ |f(x,s)| \leq A|s|^{q-1} + B, \quad \forall s \in {\mathbb R}, \mbox{ a.e. } x \in \Omega, \eqno (F_0) $$ and some positive constants $A,B$, where $1 \leq q < \frac{Np}{N-p} $ if $N \geq p+1$, and $1 \leq q < \infty$ if $1 \leq N < p$. It is well known that weak solutions $u \in W_0^{1,p}(\Omega)$ of (\ref{P}) are the critical points of the $C^1$ functional $$ \Phi(u) = \frac{1}{p} \int |\nabla u|^p \,dx- \int F(x,u) \,dx\,, $$ where $F(x,s) = \int_0^s f(x,t) \,dt$. We are interested in the situation where $\Phi$ is strongly indefinite in the sense that it is neither bounded from above or from below. Let $\lambda_{1}$ and $\lambda_{2}$ be the first and the second eigenvalues of $-\Delta_p$ on $W_0^{1p}(\Omega)$. It is known that $\lambda_{1} > 0$ is a simple eigenvalue, and that $\sigma(-\Delta_p) \cap ]\lambda_1,\lambda_2[ = \emptyset $, where $\sigma(-\Delta_p)$ is the spectrum of $-\Delta_p$, (cf. \cite {T}). We shall assume the following conditions $$ \lim_{|s| \to \infty}[f(x,s)s - pF(x,s)] = \pm \infty \quad\mbox{uniformly for a.e. } x \in \Omega\,, \eqno (F_1^{\pm}) $$ $$ \limsup_{s \to \infty} \frac{pF(x,s)}{|s|^p} < \lambda_2\,, \eqno (F_2) $$ and $$ \left[\int F(x,t\varphi _1) \,dx - \frac{1}{p}|t|^p\right] \to \infty, \quad\mbox{as } |t| \to \infty\,, \eqno (F_3) $$ where $\varphi_1$ is the normalized $\lambda_{1}$- eigenfunction. We note that $\varphi_1$ does not change sign in $\Omega$. Now, we are ready to state our main result. \begin{theorem} Assume $(F_0), (F_1^+), (F_2) $ and $(F_3)$. Then (\ref{P}) has a weak solution in $W_0^{1,p}(\Omega)$. \end{theorem} Similarly, we have \begin{theorem} Assume $(F_0), (F_1^-), (F_2)$ and $(F_3)$. Then (\ref{P}) has a weak solution in $W_0^{1,p}(\Omega)$. \end{theorem} As an immediate consequence, we obtain the following corollary. \begin{corollary} If $F$ satisfies $(F_0), (F_1^-)$, and $$ \lambda_1 \leq \liminf_{s \to \infty} \frac{pF(x,s)}{|s|^p} \leq \limsup_{s \to \infty} \frac{pF(x,s)}{|s|^p} < \lambda_2, \eqno (F'_3) $$ then (\ref{P}) has a solution. \end{corollary} The nonlinear case $(p \neq 2)$ when the nonlinearity $pF(x,s)/|s|^p$ stays asymptotically between $\lambda_1$ and $\lambda_2$ has been studied by just a few authors. A contribution in this direction is \cite {E-M}, where the authors use a topological method to study the case $N = 1$. Another contribution was made by D. G. Costa and C.A.-Magalh$\tilde {a}$es \cite{C-M} who studied the case when $pF(x,s)/|s|^p$ interacts asymptotically with the first eigenvalue $\lambda_1$. We point out, that the variational method used in the linear case $(p = 2)$ can not be extended to the nonlinear case. To overcome this difficulty, we introduce the idea of linking and proving an abstract min-max theorem. \section{Preliminaries. An abstract theorem} In this section we prove a critical-point theorem for the real functional $\Phi$ on a real Banach space $X$. Let $X^*$ denote the dual of $X$, and $\|.\|$ denote the norm in $X$ and in $X^*$. For $\Phi$ a continuously Fr\'echet differentiable map from $X$ to ${\mathbb R}$, let $\Phi'(u)$ denote its Fr\'echet derivative. For $ \Phi \in C^1(X,{\mathbb R})$ and $c \in {\mathbb R}$, let \begin{eqnarray*} &K_c = \{ x \in E: \Phi(x) = c , \Phi'(x) = 0\},&\\ &\Phi^c = \{x \in X: \Phi(u) \geq c \}.& \end{eqnarray*} Thus $K_c$ is the set of critical points of $\Phi$, and $\Phi$ has value $c$. \paragraph{Definition} Given $ c \in {\mathbb R}$, we shall say that $\Phi \in C^1(X,{\mathbb R})$ satisfies the condition $(C_c)$, if \begin{description} \item[i)] any bounded sequence $(u_n) \subset E$ such that $\Phi(u_n)\to c$ and $\Phi'(u_n) \to 0$ possesses a convergent subsequence; \item[ii)] there exist constants $\delta, R,\alpha > 0$ such that $$ \|\Phi'(u)\| \|u\| \geq \alpha {\rm \ for\ any}\ u \in\Phi^{-1}([c-\delta,c+\delta]) {\rm \ with\ } \|u\|\geq R. $$ \end{description} \paragraph{Definition} If $\Phi \in C^1(X,{\mathbb R})$ satisfies the condition $(C_c)$ for every $ c \in {\mathbb R}$, we say that $\Phi$ satisfies $(C)$. This condition was introduced by Cerami \cite {Ce}, and recently was generalized by the first author in \cite {E}. It was shown in \cite {B-B} that condition $(C)$ suffices to get a deformation lemma. \begin{lem}[Deformation Lemma] Let $X$ be a real Banach space and let $\Phi \in C^1(X,{\mathbb R})$ satisfy $(C_c)$. Then there exists $\bar {\varepsilon}>0$, $ \varepsilon \in ]0,\bar{\varepsilon}[$ and an homeomorphism $\eta : X \to X$ such that: \begin{enumerate} \item $\eta(x) = x $ if $ x \not \in \Phi^{-1}[c-\bar{\varepsilon},c+\bar{\varepsilon}[$; \item If $K_c =\emptyset $, $\eta(\Phi^{c-\varepsilon}) \subset \Phi^{c+\varepsilon}$. \end{enumerate} \end{lem} \noindent Now, we define the class of closed symmetric subsets of $X$ as $$ \Sigma = \{A \subset X: A closed, A = -A\}\,. $$ \paragraph{Definition} For a non-empty set $A$ in $\Sigma$, following Coffman \cite{C}, we define the Krasnoselskii genus as $$ \gamma(A) = \left\{ \begin{array}{l} \inf \{m : \exists h \in C(A,{\mathbb R}^m \setminus \{0\}); h(-x) = -h(x) \} \\ \infty \quad \mbox{if $\{...\}$ is empty, in particular if $0$ is in $A$.} \end{array}\right. $$ For $A$ empty we define $\gamma(A) =0$. Next we state the existence of critical points for a class of perturbations of p-homogeneous real valued $C^1$ functionals defined on a real Banach space. \begin{theorem} Let $\Phi$ be a $C^1$ functional on $X$ satisfying condition $(C)$, and let $Q$ be a closed connected subset such that $\partial Q \cap (-\partial Q) \neq \emptyset$. Assume that \begin{description} \item[i)] $\forall K \in A_2$ there exists $v_K \in K$ and there exists $\beta \in {\mathbb R}$ such that $\Phi(v_K) \geq \beta$ and $\Phi(-v_K) \geq \beta$ \item[ii)] $a = \sup_{\partial Q} \Phi < \beta$. \item[iii)] $\sup_Q \Phi(x) < \infty$. \end{description} Then $\Phi$ has a critical value $c \geq \beta$. \end{theorem} For the proof of this theorem, we will use lemma 1.1 and the following lemma. \begin{lem} Under the hypothesis of Theorem 2.1, we have $$ h(Q) \cap \Phi^\delta \not= \emptyset;\quad \forall \delta, \delta < \beta, \forall h \in \Gamma, \eqno{(H_1)}\,,$$ where $\Gamma = \{h \in C(X,X): h(x) = x \mbox{ in } \partial Q\}$. \end{lem} \paragraph{Proof :} First we claim that {\it If $A$ is nonempty connected symmetric then $\gamma(A) > 1$.} Indeed, if $\gamma(A) = 1$, then there exists a map $h$ continuous and even such that $h(A) \subset {\mathbb R} \setminus \{0\}$. Since $h$ is even continuous, $h(A)$ is a symmetric interval. Therefore, $0 \in h(A)$ which is a contradiction and the claim is proved. Let $h \in \Gamma$ and put $K = \overline{h(Q) \cup -h(Q)} $. Clearly we have $$ \partial Q \cap -\partial Q \subset h(Q) \cap -h(Q). $$ Therefore, $K$ is a closed, connected, symmetric subset, and by the claim above $ \gamma (K) \geq 2$. On the other hand, by {\bf i)} of Theorem 2.1 there exists $v_K \in K$ such that $$ \Phi(v_K) \geq \beta \quad {\rm and} \quad \Phi(-v_K) \geq \beta\,. $$ Let $\delta < \beta$, then there exists $v_1 \in h(Q) \cup -h(Q)$ such that $$ \Phi(v_1) \geq \delta \quad {\rm and} \quad \Phi(-v_1) \geq \delta\,. $$ Indeed, if this is not the case, then for every $ v \in h(Q) \cup -h(Q)$ we have $ \Phi(v) < \delta$ or $\Phi(-v) < \delta$. Then, since $\Phi$ is continuous, for every $ v \in K$ $\Phi(v) \leq \delta $ or $\Phi(-v) \leq \delta$. Which is a contradiction. Moreover, $h(Q) \cap \Phi^{\delta} \neq \emptyset$, and the conclusion easily follows. \hfill$\diamondsuit$\medskip \paragraph{ Proof of Theorem 2.1.} Suppose that $ c = \inf_{h \in \Gamma} \sup_{x \in Q}\Phi(h(x))$ is not a critical value (i.e. $ K_{c} = \emptyset)$. Let $ \bar \varepsilon < \beta - a$, then by lemma 2.1 there exists $\eta :X \to X$ an homeomorphism such that \begin{eqnarray} & \eta(x) = x \quad\mbox{if $x \not\in \Phi^{-1}[c-\bar{\varepsilon},c+\bar {\varepsilon}[$, with $\bar {\varepsilon} < \gamma - a$}; \nonumber \\ &\eta(\Phi^{c -\varepsilon}) \subset \Phi^{c + \varepsilon}\,.& \label{eq1} \end{eqnarray} By $(H_1)$ there exists a sequence $(x_n)_n \subset Q$ such that $$ \gamma \leq \sup_{n}\Phi(h(x_n)),\quad \forall h \in \Gamma\,. $$ This implies $ \beta \leq c$. Then by ${\bf iii)}$ we have $ \beta \leq c < \infty$. On the other hand, since $\bar \varepsilon < \beta - a $ and $\beta \leq c$, it results from $ii)$ that $$ \Phi(x) < c - \bar \varepsilon, \quad \forall x \in \partial Q\,. $$ This leads to \begin{equation} \eta(x) = x \quad \mbox{for $x$ in $\partial Q$}. \label{eq2} \end{equation} Hence, we have $\eta^{-1} \circ h \in \Gamma$, and by the definition of $c$ there exists $ \tilde x \in Q$ such that $$\Phi\left(\eta^{-1} \circ h(\tilde x)\right) \geq c - \varepsilon\,. $$ Hence, by (\ref{eq1}) we obtain $$ c + \varepsilon \leq \Phi\left(\eta\left[\eta^{-1} \circ h(\tilde x)\right]\right) = \Phi(h(\tilde x)). $$ Therefore, we get the contradiction $$ c + \varepsilon \leq \inf_{h \in \Gamma}\sup_{x \in Q}\Phi(h(x)) = c\,. $$ Which completes the present proof. \hfill$\diamondsuit$ \section{Proof of Theorem 1.1} In this section we shall use Theorem 2.1 for proving Theorem 1.1. The Sobolev space $W_0^{1,p}(\Omega)$ will be the Banach space $X$, endowed with the norm $\|u\| = (\int_{\Omega} |\nabla u|^p \,dx)^{\frac{1}{p}}$ and the $C^1$ functional $\Phi$ will be $$ \Phi (u) = \frac{1}{p} \int_{\Omega} |\nabla u|^p \,dx - \int_{\Omega} F(x,u) \,dx\,. $$ To apply Theorem 2.1, we shall do separate studies of the \lq\lq compactness" of $\Phi$ and its \lq\lq geometry". First, we prove that $\Phi$ satisfies the condition $(C)$. \begin{lem} Assume $F$ satisfies $ (F_0), (F_2)$ and $(F_1^+)$. Then for every $c \in {\mathbb R}$, $\Phi$ satisfies the condition $(C_c)$ on $W_0^{1,p}(\Omega)$. \end{lem} \paragraph{Proof:} We first verify the condition $(C_c)(i)$. Let $(u_n)_n \subset W_0^{1,p}(\Omega)$, be bounded and such that $\Phi '(u_n) \to 0$ in $W^{-1,p'}(\Omega)$. We have $$-\Delta _p u_n - f(x,u_n) \to 0 \quad \mbox{in } W^{-1,p'}(\Omega). $$ And as $-\Delta _p$ is an homeomorphism from $W_0^{1,p}(\Omega)$ to $W^{-1,p'}(\Omega)$ (cf \cite {Li}), we have \begin{equation} u_n - (-\Delta)_p^{-1}[f(x,u_n)] \to 0 \quad \mbox{in } W_0^{1,p}(\Omega)\,. \label{eq3} \end{equation} Since $(u_n)$ is bounded, there is a subsequence $(u'_n)$ weakly converging to some $u_0 \in W_0^{1,p}(\Omega)$. On the other hand, as the map $u \mapsto f(x,u) $ is completely continuous from $W_0^{1,p}(\Omega)$ to $W^{-1,p'}(\Omega)$ then \begin{equation} (-\Delta _p)^{-1}[f(x,u'_n) ] \to (-\Delta _p)^{-1}[f(x,u_0)] \quad \mbox{in } W_0^{1,p}(\Omega). \label{eq4} \end{equation} By (\ref{eq3}), (\ref{eq4}) we deduce that $(u'_n)$ converges in $ W_0^{1,p}(\Omega)$. Let us now prove that the condition $ (C_c)(ii)$ is satisfied for every $c \in {\mathbb R}$. Assume that $F$ satisfies $(F_0), (F_2), (F_1^+)$ and again, by contradiction, let $c \in {\mathbb R}$ and $(u_n)_n \subset W_0^{1,p}(\Omega)$ such that: \begin{eqnarray} &\Phi(u_n) \to c& \label{eq5} \\ &\|u_n\||\langle \Phi'(u_n),v\rangle | \leq \varepsilon_n \|v\| \quad \forall v \in W_0^{1,p}(\Omega) & \label{eq6} \\ & \|u_n\| \to \infty, \varepsilon_n = \|u_n\|\|\Phi'(u_n)\| \to 0 , \quad\mbox{as $n \to \infty$},&\nonumber \end{eqnarray} where $\langle .,. \rangle$ is the duality pairing between $ W_0^{1,p}(\Omega)$ and $ W^{-1,p'}(\Omega)$. It follows that $$ \lim_{n \to \infty} |\langle \Phi'(u_n), u_n\rangle - p \Phi (u_n)| = pc\,. $$ More precisely, we have \begin{equation} \lim_{n \to \infty} \int_{\Omega} \left[f(x,u_n) u_n(x) - p F(x,u_n)\right] \,dx = pc\,. \label{eq7} \end{equation} Put $z_n = u_n/\|u_n\|$, we have $\|z_n\| = 1$ and, passing if necessary to a subsequence, we may assume that: $z_n \rightharpoonup z $ weakly in $W_0^{1,p}(\Omega)$, $z_n \to z$ strongly in $L^p(\Omega)$ and $z_n(x) \to z(x) $ a.e. in $\Omega$. On the other hand, note that $\limsup_{s \to \infty} \frac{pF(x,s)}{|s|^p} < \lambda_2$ and $(F_0)$ implies \begin{equation} \label{eq8} F(x,s) \leq \frac{\lambda_2}{p} |s|^p + b(x), \quad\forall s \in {\mathbb R}, b \in L^p(\Omega)\,. \end{equation} Therefore, passing to the limit in the equality $$ \frac{1}{\|u_n\|^p} \Phi(u_n) = \frac{1}{p} - \frac{1}{\|u_n\|^p} \int F(x,u_n) \,dx $$ and, using (\ref{eq8}), it results $$ \frac{1}{p}(1 - \lambda_2 \|z\|_{L^p}^p) \leq 0 $$ which shows that $z \not\equiv 0$. Now, by $(F_1^+)$ and $(F_0)$ there exist $M > 0$, such that $$ f(x,s)s - pF(x,s) \geq -M + b_1(x), \forall s \in {\mathbb R}, \quad a.e. x \in \Omega\,; $$ hence, \begin{eqnarray} \int_{\Omega} \left[f(x,u_n) u_n(x) - p F(x,u_n)\right] \,dx &\geq& \int_{\{x: z(x) \not = 0\}} f(x,u_n) u_n(x) - p F(x,u_n) \,dx \nonumber \\ & &- M|\{x \in \Omega: z(x) = 0\}| - \|b_1\|_{L^1}.\nonumber \end{eqnarray} An application of Fatou's lemma yields $$ \int_{\Omega} \left[f(x,u_n) u_n(x) - p F(x,u_n)\right] \,dx \to \infty, \quad\mbox{as $n \to \infty$}, $$ which is a contradiction to (\ref{eq7}). Thus the proof of lemma 3.1 is complete. \hfill$\diamondsuit$\medskip Now, we will show that $\Phi$ satisfies the geometric conditions $i), ii), iii)$ of Theorem 2.1. \begin{lem} Assume that $F$ satisfies the hypothesis of Theorem 1.1. Then we have \begin{description} \item[i)] $\Phi(v) \to -\infty$, as $\|v\| \to \infty$ with $v \in X_{1}$ \item[ii)] $\forall K \in A_2$, there exists $v_K \in K$, and $\beta \in {\mathbb R}$ such that $ \Phi(v_k) \geq \beta$ and $\Phi(-v_K) \geq \beta$. \end{description} \end{lem} \paragraph{Proof:} i) Let $X_{1}$ denote the eigenspace associated to the eigenvalue $\lambda _1$. Since dim $X_{1} = 1$, we set $X_{1} = \{t\varphi _1: t \in {\mathbb R} \}$. Thus for every $v \in X_{1}, v = t\varphi_1, t \in {\mathbb R}$, we obtain \begin{eqnarray} \Phi(v) &=& \frac{1}{p}\int|\nabla t\varphi _1|^p -\int F(x,t\varphi _1) \,dx \nonumber \\ &=& \frac{1}{p}|t|^p \int|\nabla \varphi _1|^p -\int F(x,t\varphi _1) \,dx. \nonumber \end{eqnarray} Since $ \int|\nabla \varphi _1|^p = 1$, by $(F_3)$, we obtain $$ \Phi(v) = -\left[\int F(x,t\varphi _1) \,dx - \frac{1}{p}|t|^p\right] \to -\infty, \quad\mbox{as } |t| \to \infty\,. $$ ii) Let us recall that the Lusternik-Schnirelaman theory gives $$ \lambda_2 = \inf_{K \in A_2} \sup {\left\{\int |\nabla u|^p, \int |u|^p = 1, u \in K \right\}}. $$ However, for every $K \in A_2$ and $\epsilon > 0$ there exists $v_K \in K$ such that \begin{equation} (\lambda _2 - \epsilon) \int |v_K|^p \,dx \leq \int |\nabla v_K|^p \,dx\,. \label{eq9} \end{equation} Indeed, we shall treat the following two possible cases: \noindent{\bf Case 1.} $0 \in K$, (\ref{eq9}) is proved by setting $v_K = 0$. \noindent{\bf Case 2.} $0 \not\in K$, we consider $$\Pi : K \to \tilde K, v \mapsto \frac{v}{\|v\|_{L^p}}. $$ Note that $\Pi$ is an odd map. By the genus properties we have $\gamma(\Pi(K)) \geq 2$ and by the definition of $\lambda_2$ there exists $\tilde {v_K} \in \tilde K$ such that $$ \int |\tilde {v_K}|^p \,dx = 1 \quad\mbox{and}\quad (\lambda_2 - \epsilon) \leq \int |\nabla \tilde {v_K}|^p \,dx\,. $$ Thus (\ref{eq9}) is satisfied by setting $v_K = \Pi^{-1}(\tilde v_K)$. On the other hand, we note that $\limsup_{s \to \infty} \frac{pF(x,s)}{|s|^p} < \lambda_2$ and $(F_0)$ implies \begin{equation} F(x,s) \leq (\lambda_2 - 2\epsilon) \frac{|s|^p}{p} + D, \forall s \in {\mathbb R} \label{eq10} \end{equation} for some constant $D > 0$. Therefore, by using (\ref{eq9}) and (\ref{eq10}), we obtain the estimate \begin{eqnarray} \Phi(v_K) &\geq& \frac{1}{p} \int |\nabla {v_K}|^p \,dx -\frac{(\lambda_2 - 2\epsilon)}{p} \int |{v_K}|^p \,dx - D|\Omega| \nonumber \\ & \geq& \frac{1}{p} \left[1 - \frac{(\lambda_2 - 2\epsilon)}{(\lambda_2 - \epsilon)}\right] \int |\nabla {v_K}|^p \,dx - D|\Omega|\,. \label{eq11} \end{eqnarray} The argument is similar for \begin{equation} \Phi(-v_K) \geq \frac{1}{p} \left[1 - \frac{(\lambda_2 - 2\epsilon)}{(\lambda_2 - \epsilon)}\right] \int |\nabla {v_K}|^p \,dx - D|\Omega|\,. \label{eq12} \end{equation} It is clear from (\ref{eq11}) and (\ref{eq12}) that for every $K \in A_2$ we have $$ \Phi(\pm v_K) \geq - D|\Omega| = \beta.$$ Which completes the proof. \hfill$\diamondsuit$\medskip \paragraph{Proof of theorem 1.1:} In view of Lemmas 3.1 and 3.2, we may apply Theorem 2.1 letting $Q = B_R \cap X_1$, where, $ B_{R} = \{ u \in W_0^{1,p} : \|u\| \leq R\}$ with $R > 0$ being such that $\sup_{v \in \partial Q} \Phi(v) < \beta$. It follows that the functional $\Phi$ has a critical value $c \geq \beta$ and, hence, the problem (\ref{P}) has a weak solution $ u \in W_0^{1,p}(\Omega)$, the theorem is proved. \paragraph{Proof of Corollary 1.1:} The proof of this corollary follows closely the arguments in \cite {C-M}. It suffices to prove that $(F_1^-)$ and $(F'_3)$ implies $(F_3)$. Let us suppose that $g(x,s) = f(x,s) - \lambda_1 |s|^{p-1}s$ and $G(x,s) = F(x,s) - \frac{1}{p}\lambda_1 |s|^p$. Then, by $(F_1^-)$, for every $M > 0$ there exists $s_M > 0$ such that \begin{equation} g(x,s)s - pG(x,s) \leq -M, \forall |s| \geq s_M, \mbox{ a.e. }x \in \Omega\,. \label{eq13} \end{equation} Using (\ref{eq13}) and integrating the relation $$ \frac{d}{ds}\left[\frac{G(x,s)}{|s|^p}\right] = \frac{g(x,s)s - pG(x,s)}{|s|^{p+1}} $$ over an interval $[t,T] \subset [s_M,\infty[$ which was also explored in \cite {C-Ma}, we get $$ \frac{G(x,T)}{T^p} - \frac{G(x,t)}{t^p} \leq -\frac{M}{p} \left[\frac{1}{T^p} - \frac{1}{t^p}\right]. $$ Therefore, since $\liminf_{T \to \infty} \frac{G(x,T)}{T^p} \geq 0$ by $(F'_3)$, we obtain $$ G(x,t) \geq \frac{M}{p}, \forall t \geq s_M, \mbox{ a.e. }x \in \Omega $$ In the same way we show that $G(x,t) \geq \frac{M}{p},$ for every $t \leq -s_M,$ and almost every $x \in \Omega$. By $(F'_3)$ and $M > 0$ being arbitrary, we have $(F_3)$ which completes the proof. \hfill$\diamondsuit$ \begin{thebibliography}{8} \bibitem{T} A. Anane \& N. 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