\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 231--240.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{231} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Existence and uniqueness of positive solutions] {Existence and uniqueness of a positive solution for a non homogeneous problem of fourth order with weights} \author[M. Talbi, N. Tsouli \hfil EJDE/Conf/14 \hfilneg] {Mohamed Talbi, Najib Tsouli} % in alphabetical order \address{Mohamed Talbi \newline D\'epartement de Math\'ematiques et Informatique Facult\'e des Sciences, Universit\'e Mohamed 1, Oujda, Maroc} \email{talbi\_md@yahoo.fr} \address{Najib Tsouli \newline D\'epartement de Math\'ematiques et Informatique Facult\'e des Sciences, Universit\'e Mohamed 1, Oujda, Maroc} \email{tsouli@sciences.univ-oujda.ac.ma} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{35J60, 35J30, 35J65} \keywords{Ekeland's principle; p-biharmonic operator; Palais-Smale condition} \begin{abstract} In this work we study the existence of a positive solutions to the non homogeneous equation $$ \Delta( |\Delta u|^{p-2} \Delta u) = m |u|^{q-2}u $$ with Navier boundary conditions, where $1q$ and $m\in \mathcal{C}(\overline{\Omega})$, we prove the uniqueness of this solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}{Definition} \section{Introduction} We consider the following problem with Navier boundary conditions \begin{equation}\label{P1} \begin{gathered} \Delta^2_p u = m |u|^{q-2}u\quad \mbox{in } \Omega,\\ u>0\quad \mbox{in } \Omega,\\ u = \Delta u = 0 \quad \mbox{on }\partial \Omega. \end{gathered} \end{equation} Here $\Omega$ is a smooth domain in $\mathbb{R}^N$ ($N\geq 1$), $\Delta^2_p $ is the p-biharmonic operator defined by $\Delta^2_p u=\Delta (|\Delta u|^{p-2}\Delta u)$, $m\in L^{\infty}(\Omega)\setminus \{0\} , m\geq 0$ and $p, q\in ]1,p^*_2[$, $p\neq q$ where $$ p^*_2= \begin{cases} \frac{Np}{N-2p} & \mbox{if } pq$ by using the fundamental multiplicity theorem, but for $pq$ we prove the uniqueness of this solution. Notice that our approach does not use the fundamental multiplicity theorem and the mountain-pass lemma. We can refer the reader to \cite{EL} for the existence of a positive solution and to \cite{ID-OT} for the uniqueness. Similar results as ours, but with p-Laplacian operator, were studied by authors \cite{ID-OT, DI-SA}. \section{Preliminaries} In this paper, we consider the transformation of Poisson problem used by Dr\'abek and \^Otani \cite{DR-OT}. We recall some properties of the Dirichlet problem for the Poisson equation \begin{equation}\label{pbf} \begin{gathered} -\Delta u = f \quad\mbox{in } \Omega, \\ u = 0 \quad\mbox{on } \partial \Omega. \end{gathered} \end{equation} It is well known that \eqref{pbf} is uniquely solvable in $W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ for all $f\in L^p(\Omega)$ and for any $p\in ]1,+\infty[$. We denote by: $X=W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$, \\ $\|u\|_p=(\int_{\Omega}|u|^pdx)^{1/p}$ the norm in $L^p(\Omega)$,\\ $ \|u\|_{2,p}=(\|\Delta u\|_p^p+\|u\|_p^p)^{1/p}$ the norm in $X$,\\ $\|u\|_{\infty}$ the norm in $L^{\infty}(\Omega)$, \\ and $\langle\cdot,\cdot\rangle$ is the duality bracket between $L^p(\Omega)$ and $L^{p'}(\Omega)$, where $p'=p/(p-1)$. Denote by $\Lambda$ the inverse operator of $-\Delta:X\to L^p(\Omega)$. The following lemma gives us some properties of the operator $\Lambda$ (c.f. \cite{DR-OT,GI-TR}). \begin{lemma}\label{lem2.1} \begin{itemize} \item[(i)] {\rm (Continuity):} There exists a constant $c_p > 0$ such that $$ \| \Lambda f \|_{2,p} \leq c_p \|f\|_p $$ holds for all $p \in ]1, +\infty[$ and $f \in L^p (\Omega)$. \item[(ii)] {\rm (Continuity)} Given $k \in \mathbb{N}^*$, there exists a constant $c_{p,k}> 0$ such that $$ \| \Lambda f\|_{W^{k+2,p}} \leq c_{p,k} \|f\|_{W^{k,p}} $$ holds for all $p \in ]1, +\infty[$ and $f \in W^{k,p} (\Omega)$. \item[(iii)] {\rm (Symmetry)} The equality $$ \int_{\Omega} \Lambda u \cdot v dx = \int_{\Omega} u \cdot \Lambda v dx $$ holds for all $u \in L^p(\Omega)$ and $v \in L^{p'} (\Omega)$ with $p \in ]1, +\infty[$. \item[(iv)] {\rm (Regularity)} Given $f \in L^{\infty} (\Omega)$, we have $\Lambda f \in C^{1, \alpha} (\bar{\Omega})$ for all $\alpha \in ]0,1[$; moreover, there exists $c_{\alpha} > 0$ such that $$ \| \Lambda f\|_{C^{1,\alpha}} \leq c_{\alpha} \|f\|_{\infty}. $$ \item[(v)] {\rm (Regularity and Hopf-type maximum principle)} Let $f \in C (\bar{\Omega})$ and $f \geq 0$ then $w = \Lambda f \in C^{1, \alpha}(\bar{\Omega})$, for all $\alpha \in ]0,1[$ and $w$ satisfies: $w> 0$ in $\Omega, \frac{\partial w}{\partial n} < 0$ on $\partial \Omega$. \item[(vi)] {\rm (Order preserving property)} Given $f, g \in L^p (\Omega)$ if $f \leq g$ in $\Omega$, then $\Lambda f < \Lambda g$ in $\Omega$. \end{itemize} \end{lemma} Note that for all $u\in X$ and all $v \in L^p(\Omega)$, we have $v=-\Delta u$ if and only if $u=\Lambda v$. Let us denote $N_p$ the Nemytskii operator defined by \[ N_p(v)(x) = \begin{cases} |v(x)|^{p-2}v(x) & \mbox{if } v(x)\neq 0\\ 0 & \mbox{if } v(x)=0. \end{cases} \] Then for all $v \in L^p(\Omega)$ and all $w\in L^{p'}(\Omega)$, we have $N_p(v)=w$ if and only if $v=N_{p'}(w)$. For $v=-\Delta u$ which means that $u=\Lambda v$. As $X\hookrightarrow L^q(\Omega)$, then $\Lambda v\in L^q(\Omega) \forall v\in L^p(\Omega)$. We define the functionals $F,G: L^p(\Omega)\to \mathbb{R}$ as follows: $$ F(v)=\frac{1}{p}\| v\|_p^p \quad \text{and}\quad G(v)=\frac{1}{q}\int_{\Omega}m|\Lambda v|^qdx. $$ Then it is clear that $F$ and $G$ are well defined on $L^p(\Omega)$, and are of class $\mathcal{C}^1$ on $L^p(\Omega)$ and for all $v\in L^p(\Omega)$ we have $F'(v)=N_p(v)$ and $G'(v)=\Lambda (mN_q(\Lambda v))$ in $L^{p'}(\Omega)$. The operator $\Lambda$ enables us to transform problem \eqref{P1} to another problem which we shall study in the space $L^p(\Omega)$. \begin{definition} \label{def2.1} \rm We say that $u\in X\setminus \{0\}$ is a solution of problem \eqref{P1}, if $v=-\Delta u$ is a solution of the problem: Find $v\in L^p(\Omega)\setminus \{0\}$, $v>0$, such that \begin{equation} N_p(v)=\Lambda(mN_q(\Lambda v)) \quad\text{in } L^{p'}(\Omega).\label{P2} \end{equation} \end{definition} \section{Existence of a positive solution} For solutions of \eqref{P2} we understand critical points of the associated Euler-Lagrange functional $E\in \mathcal{C}^1(L^p(\Omega))$, which are given by $$ E(v)=F(v)-G(v). $$ As in \cite{DR-PO, WI}, we introduce the modified Euler-Lagrange functional defined on $\mathbb{R}\times L^p(\Omega)$ by $$ A(t,v)=E(tv). $$ If $v$ is an arbitrary element of $L^p(\Omega)$, $\partial_tA(.,v)$ (resp. $\partial_{tt}A(.,v)$)are the first (resp. second) derivative of the real valued function: $t\mapsto A(t,v)$. Since the functional $A$ is even in $t$ and that we are interested by the positive solutions, we limit our study for $t>0$. \begin{theorem}\label{solpos} Problem \eqref{P1} has a positive solution. \end{theorem} To prove theorem \ref{solpos}, we need the following preliminary results. \subsection*{Case $p>q$:} Let $v$ be an arbitrary element of $L^p(\Omega)\setminus \{0\}$. It is clair that the real valued function $t\mapsto A(t,v)$ is decreasing on $]0,t(v)[$, increasing on $]t(v),+\infty[$ and attains its unique minimum for $t=t(v)$, where \begin{equation}\label{minA} t(v)=\big(\frac{qG(v)}{pF(v)}\big)^\frac{1}{p-q}. \end{equation} On the other hand, a direct computation gives $$ A(t(v),v)=\big(\frac{1}{p}-\frac{1}{q}\big) \frac{(qG(v))^\frac{p}{p-q}}{(pF(v))^\frac{q}{p-q}}<0. $$ Furthermore we have proved in \cite{TA-TS} that $E$ is bounded bellow and coercive. We deduce that $A$ is also bounded bellow and if \begin{equation}\label{inf} \alpha =\inf_{v\in L^p(\Omega)\setminus\{0\}} A(t(v),v), \end{equation} we get $-\infty<\alpha<0$. Let $(v_n)\subset L^p(\Omega)\setminus \{0\}$ be a minimizing sequence of (\ref{inf}). Put $V_n=t(v_n)v_n$. Since $E$ is coercive the sequence $(V_n)$ is bounded. \begin{lemma}\label{liminf} The sequence $(V_n)$ satisfies $$ \liminf_{n\to +\infty}\| V_n\|_p>0. $$ \end{lemma} \begin{proof} Suppose that there is a subsequence of $(V_n)$, still denoted by $(V_n)$ such that $\lim_{n\to +\infty}\| V_n\|= 0$. It follows that $\lim_{n\to +\infty}E(V_n)= 0$; i.e. $\alpha=0$, which is impossible since $A(t(v_n),v_n)<0$. \end{proof} \begin{lemma}\label{infpos} If $\mathbb{S}$ is the unit sphere of $L^p(\Omega)$, we have $$\alpha=\inf_{v\in\mathbb{S}, v\geq 0}A(t(v),v).$$ \end{lemma} \begin{proof} For every $v\in L^p(\Omega)$, we have $|\Lambda v|\leq \Lambda |v|$ and since $p>q$, we get $$ A(t(v),v)\geq\big(\frac{1}{p}-\frac{1}{q}\big)\frac{(qG(|v|)) ^\frac{p}{p-q}}{(pF(|v|))^\frac{q}{p-q}} =A(t(|v|),|v|). $$ On the other hand the relation (\ref{minA}) implies that $\forall r>0\ and\ \forall v\in L^p(\Omega)\setminus \{0\}$, $t(v)=\frac{1}{r}t(\frac{v}{r})$. We deduce that \begin{equation}\label{infsph} \alpha =\inf_{v\in\mathbb{S}, v\geq 0}A(t(v),v), \end{equation} where $\mathbb{S}$ is the unit sphere of $L^p(\Omega)$. \end{proof} Note that the minimizing sequences considered up to here are in $\mathbb{S}$ and are nonnegative. \begin{lemma}\label{ps} Let $(v_n)\subset \mathbb{S}$ be a minimizing sequence of (\ref{infsph}), then $(V_n):=(t(v_n)v_n)$ is Palais-Smale sequence for the functional $E$. \end{lemma} \begin{proof} We have $E(V_n)\to \alpha$. We show that $$ E'(V_n)\to 0\quad\text{in } L^{p'}(\Omega). $$ Note that for every $v\in L^p(\Omega)\setminus \{0\}$, we have $\partial_tA(t(v),v)=0$ and $\partial_{tt}A(t(v),v)\neq 0$. The implicit function theorem implies that $v\to t(v)$ is $\mathcal{C}^1$ since $A$ is. Let us introduce the $\mathcal{C}^1$ functional $B$ defined on $\mathbb{S}$ by $$ B(v)=A(t(v),v)=E(t(v)v). $$ Then $$ \alpha=\inf_{v\in \mathbb{S},v\geq 0}B(v)\quad \text{and} \quad \lim_{n\to +\infty}B(v_n)=\alpha $$ Using the Ekeland variational principle on the complete manifold $(\mathbb{S},\| \cdot\|_p)$ to the functional $B$, we conclude that $$ | B'(v)(\varphi)|\leq \frac{1}{n}\|\varphi\|_p, \quad \textrm{for every } \varphi\in T_{u_n}\mathbb{S}, $$ where $T_{v_n}\mathbb{S}$ is the tangent space to $\mathbb{S}$ at the point $v_n$. Moreover, for every $\varphi\in T_{v_n}\mathbb{S}$, one has \begin{align*} B'(v_n)(\varphi) &= \partial_tA(t(v_n),v_n)t'(v_n)(\varphi)+\partial_vA(t(v),v)(\varphi) \\ &= \partial_vA(t(v),v)(\varphi), \end{align*} since $\partial_tA(t(v),v)=0$, where $t'(v)$ denotes the derivative of $v\mapsto t(v)$ at the point $v$. Furthermore, let $P: L^p(\Omega)\setminus \{0\}\to \mathbb{R}\times \mathbb{S}$, $$ v\mapsto (P_1(v),P_2(v))=(\| v\|_p,\frac{v}{\| v\|_p}). $$ Applying H\"older's inequality, for every $(v,\varphi)\in L^p(\Omega)\setminus \{0\}\times L^p(\Omega)$ we have $$ \| P'_2(v)(\varphi)\|_p\leq 2\frac{\| \varphi\|_p}{\| v\|_p}. $$ From lemma \ref{liminf} and by the fact that $\| V_n\|_p = t(v_n)$, there is a positive constant $C$ such that $$ t(v_n)\geq C, \quad \forall n\in \mathbb{N}. $$ Then for every $\varphi\in L^p(\Omega)$ we get \begin{align*} |E'(V_n)(\varphi)|&= |\partial_tA(P_1(V_n),P_2(V_n))P'_1(V_n)(\varphi) +\partial_vA(P_1(V_n),P_2(V_n))P'_2(V_n)(\varphi)| \\ &= |\partial_vA(t(v_n),v_n)P'_2(V_n)(\varphi)| \\ &= |B'(v_n)P'_2(V_n)(\varphi)| \\ &\leq \frac{1}{n}\| P'_2(V_n)(\varphi)\|_p \\ &\leq \frac{2}{n}\frac{\|\varphi\|_p}{C}. \end{align*} We easily conclude that $\lim_{n\to +\infty} E'(V_n)=0$ in $L^{p'}(\Omega)$. \end{proof} \subsection*{Case $p0$. \begin{lemma}\label{infpos2} If $\mathbb{S}$ is the unit sphere of $L^p(\Omega)$, we have $$ \alpha=\inf_{v\in\mathbb{S}, v\geq 0}A(t(v),v). $$ \end{lemma} \begin{proof} For every $v\in L^p(\Omega)\setminus \{0\}$, we have $$ A(t(v),v)=\big(\frac{1}{p}-\frac{1}{q}\big) \frac{(pF(v))^\frac{q}{q-p}}{(qG(v))^\frac{p}{q-p}}. $$ Since $|\Lambda v|\leq \Lambda|v|$, we get \[ A(t(v),v)\geq \big(\frac{1}{p}-\frac{1}{q}\big) \frac{pF(|v|)^\frac{q}{q-p}}{qG(|v|)^\frac{p}{q-p}} =A(t(|v|),|v|). \] On the other hand, the relation (\ref{maxA}) implies that for every $r>0$ and for every $v\in L^p(\Omega)\setminus\{0\}$, $t(v)=\frac{1}{r}t(\frac{v}{r})$. Hence \begin{equation}\label{infmax} \alpha=\inf_{v\in\mathbb{S}, v\geq 0}A(t(v),v). \end{equation} \end{proof} Let $(v_n)$ be a minimizing sequence of (\ref{infmax}), as in the case $p>q$, we put $$ V_n=t(v_n)v_n. $$ The proof of the following lemmas can be done like in the previous case. \begin{lemma}\label{liminf2} $\liminf_{n\to +\infty}\| V_n\|_p>0$. \end{lemma} \begin{lemma}\label{ps2} Let $(v_n)\subset \mathbb{S}$ be a minimizing sequence of (\ref{infsph}). Then $(V_n):=(t(v_n)v_n)$ is Palais-Smale sequence for the functional $E$. \end{lemma} \begin{proof}[Proof of theorem \ref{solpos}] In our paper \cite{TA-TS} we showed that $E$ verifies the Palais-Smale condition. Then by lemma \ref{ps} and lemma \ref{ps2}, we deduce that there is a subsequence of $(V_n)$, still noted by $(V_n)$ such that $V_n\to V$, $V\in L^p(\Omega)\setminus\{0\}$ and $V\geq 0$. Moreover, since $E'(V_n)\to 0$, then $E'(V)=0$. i.e. $V$ is a nonnegative solution of problem \eqref{P2}. Hence \begin{equation}\label{Vsol} N_p(V)=\Lambda (mN_q(\Lambda V)). \end{equation} The assertion (vi) of lemma \ref{lem2.1}, the relation (\ref{Vsol}) and the fact that $m\in L^p(\Omega)\setminus \{0\}$, $m\geq 0$ enable us to claim that $N_p(V)>0$ and $V>0$. Furthermore $U=\Lambda V$ is a positive solution of problem \eqref{P1}. \end{proof} \section{Uniqueness of the positive solution} \begin{theorem}\label{uniq} If $m\in \mathcal{C}(\overline{\Omega})$, $m\geq 0$ and $p>q$, then \eqref{P1} has a unique nonnegative solution. \end{theorem} Problem \eqref{P2} is equivalent to the problem: Find $v\in L^p(\Omega)\setminus \{0\}$, $v>0$ such that \begin{equation}\label{P3} N_p(v)=\| m^{1/q}\Lambda v\|_q^{q-p}\| m^{1/q}\Lambda v\|_q^{p-q}\Lambda(mN_q(\Lambda v)) \quad\text{in } L^{p'}(\Omega). \end{equation} To prove that problem \eqref{P2} has a unique nonnegative solution, we will study the principal positive eigenvalue of the eigenvalue problem: Find $v\in L^p(\Omega)\setminus \{0\}\times \mathbb{R_+^*}$ such that \begin{equation}\label{VP} N_p(v)=\lambda\| m^{1/q}\Lambda v\|_q^{p-q}\Lambda(mN_q(\Lambda v)) \quad\textrm{in}\quad L^{p'}(\Omega). \end{equation} Consider the functionals $f$ and $g$ defined on $L^p(\Omega)$ by $$ f(v)=\frac{1}{p}\| v\|_p\quad \text{and}\quad g(v)=\frac{1}{p}(\int_{\Omega}m|\Lambda v|^qdx)^\frac{p}{q}. $$ Hence problem (\ref{VP}) is equivalent to the problem: Find $(v,\lambda)\in L^p(\Omega)\setminus \{0\}\times \mathbb{R^*_+}$ such that \begin{equation} f'(v)=\lambda g'(v)\quad\textrm{in } L^{p'}(\Omega). \end{equation} Define $$ \lambda_1=\inf _{v\in M}f(v), $$ where $M= \{v\in L^p(\Omega)/g(v)=1\}$. We need the preliminary results. \begin{lemma}\label{vp1} \begin{itemize} \item[(i)] $\lambda_1$ is the first positive eigenvalue of problem (\ref{VP}). Moreover $v_1$ is an eigenfunction associated with $\lambda_1$ if and only if $$ f(v_1)-\lambda _1g(v_1)=0=\inf_{v\in L^p(\Omega)\setminus \{0\}}f(v)-\lambda_1g(v). $$ \item[(ii)] Every eigenfunction associated with $\lambda_1$ is positive or negative. \end{itemize}\end{lemma} \begin{proof} (i) The functional $f$ is weakly semi-continuous below and coercive on $M$. Since $g$ is weakly continuous, then $M$ is weakly closed. Hence there is $v_1\in M$ such that $f(v_1)=\lambda_1=\lambda_1g(v_1)$. The p-homogeneity of $f$ and $g$ implies that $\lambda_1$ is an eigenvalue of problem (\ref{VP}) if and only if $$ \forall v\in L^p(\Omega)\setminus\{0\},\quad \lambda_1\leq \frac{f(v)}{|g(v)|} $$ if and only if for all $v\in L^p(\Omega)\setminus \{0\}$, \[ f(v)-\lambda_1 g(v)\geq f(v)-\lambda_1 |g(v)| \geq 0= f(v_1)-\lambda_1 g(v_1). \] Now we show that $\lambda_1$ is the first positive eigenvalue: Suppose on the contrary that there exits $\lambda\in ]0, \lambda_1[$ and $v\in L^p(\Omega)\setminus\{0\}$ such that $f(v)-\lambda g(v)=0$. Then we get $$ 0=f(v_1)-\lambda_1g(v_1)\leq f(v)-\lambda_1g(v)0$ in $\Omega$. Hence $v$ is positive or negative in $\Omega$.\end{proof} \begin{lemma}\label{maxmin} If $v$ and $w$ are positive eigenfunctions of \eqref{P2} associated with $\lambda_1$, then the functions $\max$ and $\min$ defined in $\Omega$ by $\max(x) = \max(v(x),w(x))$ and $\min(x) = \min(u(x),w(x))$ are also solutions of \eqref{P2} associated with $\lambda_1$. \end{lemma} To prove lemma \ref{maxmin} we need the following results. \begin{lemma}\label{dr-ot} Let $a, b, c$ and $p$ be reals such that $a\geq 0$, $b\geq 0$ and $p>1$. If $c\geq \max\{b-a, 0\}$, then $$ |a+c|^p+|b-c|^p\geq a^p+b^p. $$ \end{lemma} For the proof of the above lemma see for example \cite{DR-OT}. \begin{lemma}\label{mh} Let $a, b, c$ and $d$ be in $\mathbb{R}_+$ such that $a\geq \max(c,d)$. If $a+b\geq c+d$, then for every $p\in [1,+\infty[$, $ a^p+b^p\geq c^p+d^p$. \end{lemma} \begin{proof} If $b\geq \min(c,d)$ or $a\geq c+d$ it is evident. Else, set $\alpha =a-d$ and $\beta = c-b$. We can suppose that $d\leq c$. Since $a< c+d$ and $a+b\geq c+d$ we deduce that $\alpha0$ and $t<0$ and letting $t$ tend to $0^{\pm}$, we get $$ \nabla N_p(v)(x_0)=\nabla N_p(\max{}_k )(x_0) =k^{p-1} \nabla N_p(w)(x_0). $$ Thus \begin{align*} \nabla \big(\frac{ N_p(v)}{ N_p(w)}\big)(x_0) &=\nabla (\frac{N_p(v)}{N_p(w)})(x_0)\\ &=\frac{(\nabla (N_p(v))(x_0) N_p(w)(x_0)- N_p(v)(x_0)\nabla( N_p(w))(x_0))}{(N_p(w)(x_0))^2}= 0. \end{align*} Hence $$ N_p(\frac{v}{w})=\frac{N_p(v)}{N_p(w)}=\text{\rm const} =k^{p-1}\quad \text{in }\Omega $$ and $$ \frac{v}{w}=k\quad \text{in } \Omega. $$ Now we show that every positive eigenfunction is associated with $\lambda_1$: Let $\lambda>\lambda_1$, suppose that problem \eqref{P2} has a positive eigenfunction $w$ associated with $\lambda$ and let $v$ be a positive solution of problem \eqref{P2} associated with $\lambda_1$, we have $$ N_p(v)=\lambda_1\Lambda(mN_p(\Lambda v))\quad\text{and}\quad N_p(w)=\lambda\Lambda(mN_p(\Lambda w)). $$ Then from the assertion in lemma \ref{lem2.1}(v) we deduce that $ N_p(v)$ and $N_p(w)$ are in $\mathcal{C}^{1,\alpha}(\overline{\Omega})$, and $$ \partial( N_p(v))/\partial n<0,\quad\partial(N_p(w))/\partial n < 0\quad \text{on }\partial \Omega. $$ It follows that $N_p(v)/N_p(w)$ is in $\mathcal{C}(\overline{\Omega})$. Set $$ a=\max_{x\in\overline{\Omega}}N_p(v)(x)/N_p(w)(x). $$ We deduce that $N_p(v)\leq aN_p(w)$. The monotonicity of $N_{p'}$ implies $$ v\leq a^{\frac{1}{p-1}}w. $$ Since problem \eqref{P2} is homogeneous, $a^{\frac{1}{p-1}}w$ is also a solution of problem \eqref{P2}, we may assume without loss of generality that $v\leq w$. Then, from the assertion of lemma \ref{lem2.1}(vi) and by the monotonicity of $N_q$, we get \begin{align*} N_p(v)&=\lambda_1\| m^{1/q} \Lambda v\|_q^{p-q}\Lambda(mN_q(\Lambda v))\\ &\leq \| m^{1/q} \Lambda w\|_q ^{p-q}\lambda_1\Lambda(mN_q(\Lambda w))\\ &=\lambda\| m^{1/q} \Lambda cw\|_q ^{p-q}\Lambda(mN_q(\Lambda cw))\\ &= N_p(cw), \end{align*} where $$ c=(\lambda_1/\lambda)^{1/(p-1)}<1. $$ Hence it follows by the monotonicity of $N_{p'}$ that $v0$ such that $w=kv$. It follows that $v=w$. \end{proof} \begin{thebibliography}{00} \bibitem{AD} R. A. Adams, \emph{Sobolev Spaces,} Academic Press, New York, (1975). \bibitem{DI-SA} J. I. Diaz, J. E. 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