\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 134, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/134\hfil Steklov problems] {Steklov problems involving the $p(x)$-Laplacian} \author[G. A. Afrouzi, A. Hadjian, S. Heidarkhani \hfil EJDE-2014/134\hfilneg] {Ghasem A. Afrouzi, Armin Hadjian, Shapour Heidarkhani} % in alphabetical order \address{Ghasem A. Afrouzi \newline Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran} \email{afrouzi@umz.ac.ir} \address{Armin Hadjian \newline Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran} \email{hadjian83@gmail.com} \address{Shapour Heidarkhani \newline Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran} \email{s.heidarkhani@razi.ac.ir} \thanks{Submitted December 26, 2013. Published June 10, 2014.} \subjclass[2000]{35J60, 35J20} \keywords{$p(x)$-Laplace operator; variable exponent Sobolev spaces;\hfill\break\indent multiple solutions; variational methods} \begin{abstract} Under suitable assumptions on the potential of the nonlinearity, we study the existence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian. Our approach is based on variational methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The aim of this article is to study the following Steklov problem involving the $p(x)$-Laplacian, \begin{equation}\label{e1.1} \begin{gathered} \Delta_{p(x)} u=a(x)|u|^{p(x)-2}u \quad \text{in } \Omega,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda f(x,u) \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, $\lambda$ is a positive parameter, $p\in C(\bar{\Omega})$, $\Delta_{p(x)}u:=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)$ denotes the $p(x)$-Laplace operator, $f:\partial\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory function, $a\in L^\infty(\Omega)$ with $\operatorname{ess\,inf}_{\Omega}a>0$ and $\nu$ is the outer unit normal to $\partial\Omega$. The study of differential equations and variational problems with nonstandard $p(x)$-growth conditions is a new and interesting topic. It varies from nonlinear elasticity theory, electro-rheological fluids, and so on (see \cite{Ru,Zh}). Many results have been obtained on this kind of problems, for instance we here cite \cite{AlElOu, BonaChi1, BonaChi2, BonaChi3, CamChDi, DagSci, Dai, FanDeng, FanHan, FanZhang, HarHaLat, Ji}. The inhomogeneous Steklov problems involving the $p$-Laplacian has been the object of study in, for example, \cite{MavNka}, in which the authors have studied this class of inhomogeneous Steklov problems in the cases of $p(x)\equiv p=2$ and of $p(x)\equiv p> 1$, respectively. In this paper, motivated by \cite{AlElOu}, at first, we prove the existence of a non-zero solution of the problem \eqref{e1.1}, without assuming any asymptotic condition neither at zero nor at infinity (see Theorem \ref{the3.1}). Next, we obtain the existence of two solutions, possibly both non-zero, assuming only the classical Ambrosetti-Rabinowitz condition; that is, without requiring that the potential $F$ satisfies the usual condition at zero (see Theorems \ref{the3.2} and \ref{the3.3}). Finally, we present a three solutions existence result under appropriate condition on the potential $F$ (see Theorem \ref{the3.4}). Our approach is fully variational method and the main tools are critical point theorems contained in \cite{Bonanno2} and \cite{BonaMara} (see Theorems \ref{the2.1} and \ref{the2.2} in the next section). A special case of Theorem \ref{the3.4} is the following theorem. \begin{theorem}\label{t1.1} Let $p(x)=p>N$ for every $x\in\Omega$ and let $f:\mathbb{R}\to\mathbb{R}$ be a non-negative continuous function. Put $F(t):=\int_0^tf(\xi)d\xi$ for each $t\in \mathbb{R}$. Assume that $F(d)>0$ for some $d\geq 1$ and, moreover, $$ \liminf_{\xi\to 0}\frac{F(\xi)}{\xi^{p}}=\limsup_{|\xi|\to+\infty}\frac{F(\xi)}{\xi^{p}}=0. $$ Then, there is $\lambda^\star>0$ such that for each $\lambda>\lambda^\star$ the problem \begin{gather*} \Delta_p u=a(x)|u|^{p-2}u \quad \text{in } \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda f(u) \quad \text{on } \partial\Omega, \end{gather*} admits at least three non-negative weak solutions. \end{theorem} \section{Preliminaries} In this section, we recall definitions and theorems to be used in this paper. Let $(X,\|\cdot\|)$ be a real Banach space and $\Phi$, $\Psi:X\to \mathbb{R}$ be two continuously G\^ateaux differentiable functionals; put $$ I:=\Phi-\Psi $$ and fix $r_1$, $r_2\in[-\infty,+\infty]$, with $r_1 0 : \int_\Omega\big|\frac{u(x)}{\lambda}\big|^{p(x)}dx\leq 1 \big\}. $$ Define the variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ by $$ W^{1,p(x)}(\Omega):=\big\{u\in L^{p(x)}(\Omega) : |\nabla u|\in L^{p(x)}(\Omega)\big\} $$ equipped with the norm $$ \|u\|_{W^{1,p(x)}(\Omega)}:=|u|_{p(x)}+|\nabla u|_{p(x)}. $$ It is well known \cite{FanZhao} that, in view of \eqref{e2.2}, both $L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$, with the respective norms, are separable, reflexive and uniformly convex Banach spaces. When $a\in L^\infty(\Omega)$ with $\operatorname{ess\,inf}_{\Omega}a>0$, for any $u\in W^{1,p(x)}(\Omega)$, define $$ \|u\|_{a}:=\inf \Big\{\lambda > 0 : \int_\Omega \big(|\frac{\nabla u(x)}{\lambda}|^{p(x)} + a(x)|\frac{u(x)}{\lambda}|^{p(x)}\big)dx\leq 1 \Big\}. $$ Then, it is easy to see that $\|u\|_{a}$ is a norm on $W^{1,p(x)}(\Omega)$ equivalent to $\|u\|_{W^{1,p(x)}(\Omega)}$. In the following, we will use $\|u\|_{a}$ instead of $\|u\|_{W^{1,p(x)}(\Omega)}$ on $X=W^{1,p(x)}(\Omega)$. As pointed out in \cite{KovRak} and \cite{FanZhao}, $X$ is continuously embedded in $W^{1,p^-}(\Omega)$ and, since $p^->N$, $W^{1,p^-}(\Omega)$ is compactly embedded in $C^0(\bar{\Omega})$. Thus, $X$ is compactly embedded in $C^0(\bar{\Omega})$. So, in particular, there exists a positive constant $m>0$ such that \begin{equation}\label{e2.3} \|u\|_{C^0(\bar{\Omega})}\leq m\|u\|_a \end{equation} for each $u\in X$. When $\Omega$ is convex, an explicit upper bound for the constant $m$ is $$ m\leq 2^{\frac{p^{-}-1}{p^{-}}}\max\Big\{\big(\frac{1}{\|a\|_{1}}\big)^{\frac{1}{p^{-}}}, \frac{d}{N^{\frac{1}{p^{-}}}} \big(\frac{p^{-}-1}{p^{-}-N}|\Omega|\big)^{\frac{p^{-}-1}{p^{-}}} \frac{\|a\|_{\infty}}{\|a\|_{1}}\Big\}\big(1+|\Omega|\big) $$ where $d:=\operatorname{diam}(\Omega)$ and $|\Omega|$ is the Lebesgue measure of $\Omega$ (for details, see \cite{DagSci}), $\|a\|_1:=\int_\Omega a(x)dx$ and $\|a\|_{\infty}:=\sup_{x\in \Omega}a(x)$. \begin{lemma}[{\cite{FanZhao}}]\label{lem2.3} Let $I(u)=\int_{\Omega}(|\nabla u|^{p(x)}+a(x)|u|^{p(x)})dx$. For $u\in X$ we have \begin{itemize} \item[(i)] $\|u\|_a<1(=1;>1)\Leftrightarrow I(u)<1(=1;>1);$ \item[(ii)] If $\|u\|_a<1\Rightarrow \|u\|_a^{p^{+}}\leq I(u)\leq \|u\|_a^{p^{-}};$ \item[(iii)] If $\|u\|_a>1\Rightarrow \|u\|_a^{p^{-}}\leq I(u)\leq \|u\|_a^{p^{+}}$. \end{itemize} \end{lemma} We refer the reader to \cite{FanShenZhao, FanZhao} for the basic properties of the variable exponent Lebesgue and Sobolev spaces. Throughout this article, we assume the following condition on the Carath\'{e}odory function $f:\partial\Omega\times\mathbb{R}\to\mathbb{R}$: \begin{itemize} \item[(F0)] $|f(x,s)|\leq \alpha(x)+b|s|^{\beta(x)-1}$ for all $(x,s)\in \partial\Omega\times\mathbb{R}$, where $\alpha\in L^{\frac{\beta(x)}{\beta(x)-1}}(\partial\Omega)$, $b\geq 0$ is a constant and $\beta\in C(\partial\Omega)$ such that \begin{equation}\label{e2.4} 1<\beta^{-}:=\inf_{x\in\bar{\Omega}}\beta(x)\leq\beta(x)\leq \beta^{+}:=\sup_{x\in\bar{\Omega}}\beta(x)p^+$ and $R>0$ such that for all $x\in\partial\Omega$ and $|s|\geq R$, $$ 0<\mu F(x,s)\leq s f(x,s). $$ \end{itemize} Then, for each $\lambda\in\Lambda$, where $\Lambda$ is given by \eqref{e3.2}, the problem \eqref{e1.1} has at least two non-trivial weak solutions $\bar{u}_{1},\ \bar{u}_{2}\in X$ such that $$ \max_{x\in\Omega}|\bar{u}_{1}(x)|0$ such that $\inf_{\|u-\bar{u}_1\|=\rho}I_\lambda(u)>I_\lambda(\bar{u}_1)$, so condition \cite[$(I_1)$, Theorem 2.2]{Rab} is verified. From (AR), by standard computations, there is a positive constant $C$ such that \begin{equation}\label{e3.3} F(x,s)\geq C|s|^\mu \end{equation} for all $x\in \partial\Omega$ and $|s|>R$. In fact, setting $\gamma(x)=\min_{|\xi|=R}F(x,\xi) $ and \begin{equation}\label{e3.4} \varphi_s(t)=F(x,ts)\quad \forall t>0, \end{equation} by (AR), for every $ x\in \partial\Omega$ and $|s|>R$ one has $$ 0<\mu\varphi_s(t)=\mu F(x,ts)\leq ts f(x,ts)=t\varphi'_s(t)\quad \forall t>0. $$ Therefore, $$ \int_{R/|s|}^1\frac{\varphi'_s(t)}{\varphi_s(t)}dt\geq \int_{R/|s|}^1\frac{\mu}{t}dt. $$ Then $$ \varphi_s(1)\geq \varphi_s\Big(\frac{R}{|s|}\Big)|s|^\mu. $$ Taking into account \eqref{e3.4}, we obtain $$ F(x,s)\geq F\Big(x,\frac{R}{|s|}s\Big)|s|^\mu\geq \gamma(x)|s|^\mu\geq C|s|^\mu, $$ and \eqref{e3.3} is proved. Now, by choosing any $u\in X\setminus\{0\}$ and $t>1$, one has \begin{align*} I_\lambda(tu)&=(\Phi-\lambda\Psi)(tu)\\ &=\int_\Omega\frac{t^{p(x)}}{p(x)}\left(|\nabla u|^{p(x)}+a(x)|u|^{p(x)}\right)dx -\lambda\int_{\partial\Omega}F(x,tu(x))d\sigma\\ &\leq t^{p^+}\int_\Omega\frac{1}{p(x)}\left(|\nabla u|^{p(x)}+a(x)|u|^{p(x)}\right)dx -Ct^\mu\lambda\int_{\partial\Omega}|u(x)|^\mu d\sigma. \end{align*} Since $\mu>p^+$, the functional $I_\lambda$ is unbounded from below. So, condition \cite[$(I_2)$, Theorem 2.2]{Rab} is verified. Therefore, $I_\lambda$ satisfies the geometry of mountain pass. Now, to verify the (PS)-condition it is sufficient to prove that any (PS)-sequence is bounded. To this end, suppose that $\{u_{n}\}\subset X$ is a (PS)-sequence; i.e., there is $M>0$ such that $$ \sup|I_\lambda(u_{n})|\leq M,\quad I'_\lambda(u_{n})\to 0\quad\text{ as } n\to +\infty. $$ Let us show that $\{u_{n}\}$ is bounded in $X$. Using hypothesis (AR), since $I_\lambda(u_n) $ is bounded, we have for $n$ large enough: \begin{align*} M+1 &\geq I_\lambda(u_n)-\frac{1}{\mu}\langle I'_\lambda(u_n),u_n\rangle +\frac{1}{\mu}\langle I'_\lambda(u_n),u_n\rangle\\ &=\int_{\Omega}\frac{1}{p(x)}\left(|\nabla u_n|^{p(x)}+a(x)|u_n|^{p(x)}\right)dx -\lambda\int_{\partial\Omega}F(x,u_n(x))d\sigma\\ &\quad-\frac{1}{\mu}\Big[\int_{\Omega}\left(|\nabla u_n|^{p(x)}+a(x)|u_n|^{p(x)}\right)dx-\lambda\int_{\partial\Omega}f(x,u_n(x))u_n(x) d\sigma\Big]\\ &\quad+\frac{1}{\mu}\langle I'_\lambda(u_n),u_n\rangle\\ &\geq\big(\frac{1}{p^{+}}-\frac{1}{\mu}\big)\|u_n\|_a^{p^{-}}-\frac{1}{\mu} \|I'_\lambda(u_n)\|_{X^\ast}\|u_n\|_a-c_1\\ &\geq\big(\frac{1}{p^{+}}-\frac{1}{\mu}\big)\|u_n\|_a^{p^{-}} -\frac{c_2}{\mu}\|u_n\|_a-c_1, \end{align*} where $c_1$ and $c_2$ are two positive constants. Since $\mu>p^+$, from the above inequality we know that $\{u_n\}$ is bounded in $X$. Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point $\bar{u}_2$ of $I_\lambda$ such that $I_\lambda(\bar{u}_2)>I_\lambda(\bar{u}_1)$. So, $\bar{u}_1$ and $\bar{u}_2$ are two distinct weak solutions of \eqref{e1.1} and the proof is complete. \end{proof} Here we give the following result as a direct consequence of Theorem \ref{the3.2} in the autonomous case. \begin{theorem}\label{the3.3} Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $f(0)\neq 0$ and $|f(s)|\leq \alpha+b|s|^{\beta-1}$ for all $s\in \mathbb{R}$, where $\alpha>0$, $b\geq 0$ and $1<\betap^+$ and $R>0$ such that for all $|s|\geq R$, $$0<\mu F(s)\leq sf(s), $$ \end{itemize} and for each $$ \lambda\in\Big]\frac{d^{p^+}\|a\|_1}{p^-|\partial\Omega| F(d)},\frac{\left(\frac{c}{m}\right)^{p^-}}{p^+|\partial\Omega|\max_{|t|\leq c}F(t)}\Big[, $$ the problem \begin{gather*} \Delta_{p(x)} u=a(x)|u|^{p(x)-2}u \quad \text{in } \Omega,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda f(u) \quad \text{on } \partial\Omega, \end{gather*} admits at least two non-trivial weak solutions $\bar{u}_{1},\bar{u}_{2}\in X$ such that $$ \max_{x\in\Omega}|\bar{u}_{1}(x)|\left(\frac{c}{m}\right)^{p^-}$, such that the assumption \eqref{e3.1} in Theorem \ref{the3.1} holds. Then, for each $\lambda\in\Lambda$, where $\Lambda$ is given by \eqref{e3.2}, the problem \eqref{e1.1} has at least three weak solutions. \end{theorem} \begin{proof} Our goal is to apply Theorem \ref{the2.2}. The functionals $\Phi$ and $\Psi$ defined in the proof of Theorem \ref{the3.1} satisfy all regularity assumptions requested in Theorem \ref{the2.2}. So, our aim is to verify (i) and (ii). Arguing as in the proof of Theorem \ref{the3.1}, put $r:=\frac{1}{p^+}\big(\frac{c}{m}\big)^{p^-}$ and $w(x):=d$ for all $x\in\bar{\Omega}$, bearing in mind that $d^{p^-}\|a\|_1>(\frac{c}{m})^{p^-}$, we have $$ \Phi(w)=\int_\Omega\frac{1}{p(x)}a(x)d^{p(x)}dx \geq\frac{1}{p^+}d^{p^-}\|a\|_1>r>0. $$ Therefore, the assumption (i) of Theorem \ref{the2.2} is satisfied. We prove that the functional $I_\lambda$ is coercive for all $\lambda>0$. If $u\in X$, then by condition \eqref{e2.4} and the embedding theorem (see \cite[Theorem 2.1]{Deng}) we have $u\in L^{\beta(x)}(\partial\Omega)$. Then there is some constant $C>0$ such that $$ \|u\|_{L^{\beta(x)}(\partial\Omega)}\leq C\|u\|_a,\quad\forall u\in X. $$ Now, by using H\"{o}lder inequality (see \cite{FanZhao}) and condition {\rm (F0)}, for all $u\in X$ such that $\|u\|_a\geq 1$, we have \begin{align*} \Psi(u)&=\int_{\partial\Omega}F(x,u(x))d\sigma =\int_{\partial\Omega}\Big(\int_{0}^{u(x)}f(x,t)dt\Big)d\sigma\\ &\leq \int_{\partial\Omega}\Big(\alpha(x)|u(x)|+\frac{b}{\beta(x)}|u(x)|^{\beta(x)}\Big)d\sigma \\ &\leq 2\|\alpha\|_{L^{\frac{\beta(x)}{\beta(x)-1}}(\partial\Omega)} \|u\|_{L^{\beta(x)}(\partial\Omega)} +\frac{b}{\beta^{-}}\int_{\partial\Omega}|u(x)|^{\beta(x)}d\sigma\\ &\leq 2C\|\alpha\|_{L^{\frac{\beta(x)}{\beta(x)-1}}(\partial\Omega)} \|u\|_a+\frac{b}{\beta^{-}}\int_{\partial\Omega}|u(x)|^{\beta(x)}d\sigma. \end{align*} On the other hand, there is a constant $C'>0$ such that \[ \int_{\partial\Omega}|u(x)|^{\beta(x)}d\sigma\leq \max\left\{\|u\|_{L^{\beta(x)}(\partial\Omega)}^{\beta^{+}}, \|u\|_{L^{\beta(x)}(\partial\Omega)}^{\beta^{-}}\right\}\leq C'\|u\|_a^{\beta^{+}}. \] Then, $$ \Psi(u)\leq 2C\|\alpha\|_{L^{\frac{\beta(x)}{\beta(x)-1}}(\partial\Omega)} \|u\|_a+\frac{b}{\beta^{-}}C'\|u\|_a^{\beta^{+}}. $$ Since \[ \Phi(u)=\int_{\Omega}\frac{1}{p(x)}\left(|\nabla u|^{p(x)}+a(x)|u|^{p(x)}\right)dx\geq\frac{1}{p^{+}}\|u\|_a^{p^{-}}, \] for every $\lambda>0$ we have $$ I_\lambda(u)\geq\frac{1}{p^{+}}\|u\|_a^{p^{-}}-2\lambda C\|\alpha\|_{L^{\frac{\beta(x)}{\beta(x)-1}}(\partial\Omega)}\|u\|_a -\frac{\lambda b C'}{\beta^{-}}\|u\|_a^{\beta^{+}}. $$ Since $p^{-}>\beta^{+}$, the functional $I_\lambda$ is coercive. Then also condition (ii) holds. So, for each $\lambda\in\Lambda$, the functional $I_\lambda$ admits at least three distinct critical points that are weak solutions of problem \eqref{e1.1}. \end{proof} \begin{remark}\label{rem3.5}\rm{ If we assume that $f:\partial\Omega\times\mathbb{R}\to\mathbb{R}$ is a non-negative Carath\'{e}odory function satisfying (F0), then the previous theorems guarantee the existence of non-negative weak solutions. In fact, let $\bar{u}$ be a weak solution of the problem \eqref{e1.1}. We claim that it is non-negative. Arguing by contradiction and setting $A:=\{x\in\bar{\Omega} : \bar{u}(x)<0\}$, one has $A\neq\emptyset$. Put $\bar{v}:=\min\{\bar{u},0\}$, one has $\bar{v}\in X$. So, taking into account that $\bar{u}$ is a weak solution and by choosing $v=\bar{v}$, one has $$ \int_A|\nabla\bar{u}|^{p(x)}dx+\int_A a(x)|\bar{u}|^{p(x)}dx=\lambda\int_{\partial\Omega} f(x,\bar{u}(x))\bar{u}(x)d\sigma\leq 0, $$ that is, $\|\bar{u}\|_{W^{1,p(x)}(A)}=0$ which is absurd. Hence, our claim is proved. Also, when $f$ is a non-negative function, condition \eqref{e3.1} becomes $$ \frac{\int_{\partial\Omega}F(x,c)d\sigma}{\left(\frac{c}{m}\right)^{p^-}} <\frac{p^-\int_{\partial\Omega}F(x,d)d\sigma}{p^+d^{p^+}\|a\|_1}. $$ In this case, the previous theorems ensure the existence of non-negative solutions to the problem \eqref{e1.1} for each $$ \lambda\in\Big]\frac{d^{p^+}\|a\|_1}{p^-\int_{\partial\Omega} F(x,d)d\sigma},\frac{\left(\frac{c}{m}\right)^{p^-}}{p^+\int_{\partial\Omega}F(x,c)d\sigma}\Big[. $$} \end{remark} \begin{remark}\label{rem3.6}\rm{ Theorems \ref{the3.1} and \ref{the3.4} ensure more precise conclusions rather than \cite[Theorems 1.1 and 1.3]{AlElOu}. In fact, Theorem 1.1 of \cite{AlElOu} proves that for any $\lambda\in]0,+\infty[$, the problem \eqref{e1.1}, when $a\equiv 1$, has at least a non-trivial weak solution. Also, Theorem 3.1 of \cite{AlElOu} establishes that there exists an open interval $\Lambda\subset]0,+\infty[$ such that, for every $\lambda\in\Lambda$, the problem \eqref{e1.1}, when $a\equiv 1$, admits at least three solutions. Hence, a location of the interval $\Lambda$ in $]0,+\infty[$ is not established.} \end{remark} \begin{proof}[Proof of Theorem \ref{t1.1}] Fix $\lambda>\lambda^\star:=\frac{d^p\|a\|_1}{p|\partial\Omega| F(d)}$ for some $d\geq 1$ such that $F(d)>0$. Since $$ \liminf_{\xi\to 0}\frac{F(\xi)}{\xi^{p}}=0, $$ there is a sequence $\{c_n\}\subset]0,+\infty[$ such that $\lim_{n\to +\infty} c_{n}=0$ and $$ \lim _{n\to +\infty}\frac{F(c_n)}{c_n^{p}}=0. $$ Therefore, there exists $\overline{c}\geq m$ such that $$ \frac{F(\overline{c})}{\overline{c}^{p}}< \min\big\{\frac{F(d)}{(md)^{p}\|a\|_{1}},\frac{1}{p|\partial\Omega|m^{p} \lambda}\big\} $$ and $\overline{c}