\documentclass[twoside]{article} \usepackage{amsfonts, amsmath, amssymb, epic} \pagestyle{myheadings} \markboth{Geometry of the energy functional and the Fredholm alternative} { Pavel Dr\'abek} \begin{document} \setcounter{page}{103} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems,\newline Electronic Journal of Differential Equations, Conference 08, 2002, pp 103--120. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Geometry of the energy functional and the Fredholm alternative for the $p$-Laplacian in higher dimensions % \thanks{ {\em Mathematics Subject Classifications:} 35J60, 35P30, 35B35, 49N10. \hfil\break\indent {\em Key words:} $p$-Laplacian, variational methods, PS condition, Fredholm alternative, \hfil\break\indent upper and lower solutions \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published October 21, 2002.} } \date{} \author{Pavel Dr\'abek} \maketitle \begin{abstract} In this paper we study Dirichlet boundary-value problems, for the $p$-Laplacian, of the form $$\displaylines{ - \Delta_p u -\lambda_1 |u|^{p-2} u = f\quad\mbox{ in }\Omega,\cr u = 0 \quad\mbox{ on } \partial \Omega, }$$ where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N \geq 1, p>1$, $f \in C (\bar{\Omega})$ and $\lambda_1 > 0$ is the first eigenvalue of $\Delta_p$. We study the geometry of the energy functional $$ E_p(u) = \frac{1}{p} \int_{\Omega} |\nabla u|^p - \frac{\lambda_1}{p} \int_{\Omega} |u|^p - \int_{\Omega} f u $$ and show the difference between the case $12$. We also give the characterization of the right hand sides $f$ for which the Dirichlet problem above is solvable and has multiple solutions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \numberwithin{equation}{section} \section {Introduction and statement of the results} Our aim is to study the solvability of the Dirichlet boundary-value problem \begin{equation}\label{1.1} \begin{gathered} -\Delta_p u - \lambda_1 |u|^{p-2} u = f \quad\mbox{in }\Omega,\\ u = 0 \quad \mbox{on }\partial \Omega. \end{gathered} \end{equation} Here $p>1$ is a real number, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with sufficiently smooth boundary $\partial \Omega$, $\Delta_p u = \mathop{\rm div} (|\nabla u|^{p-2} \nabla u)$ is the $p$-Laplacian and $f\in C(\bar \Omega)$. We assume that if $N\geq 2$ then $\partial \Omega$ is a compact connected manifold of class $C^2$. By $\lambda_1$ we denote the first eigenvalue of the related homogeneous eigenvalue problem \begin{equation}\label{1.2} \begin{gathered} -\Delta_p u - \lambda |u|^{p-2} u = 0 \quad\mbox{in }\Omega, \\ u=0 \quad\mbox{on } \partial \Omega. \end{gathered} \end{equation} In this paper, the function $u$ is said to be a ({\it weak}) {\it solution} of (\ref{1.1}) if $u \in W^{1,2}_0 (\Omega)$ and the integral identity \begin{equation}\label{1.3} \int_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v - \lambda_1 \int_{\Omega} |u|^{p-2} u v = \int_{\Omega} f v \end{equation} holds for all $v \in W^{1,p}_0 (\Omega)$. As for the properties of $\lambda_1$ (see e.g. \cite{2,17}), let us mention that $\lambda_1$ is positive, simple and isolated and the corresponding eigenfunction $\varphi_1$ (associated with $\lambda_1$) satisfies $\varphi_1 > 0$ in $\Omega, \frac{\partial \varphi_1}{\partial n} < 0$ on $\partial \Omega$, where $n$ denotes the exterior unit normal to $\partial \Omega$. One also has $\varphi_1 \in C^{1,\nu} (\bar \Omega)$ with some $\nu \in (0, 1)$ (see e.g. [9, Lemma \ref{lem2.1}, p. 115]). Moreover, $\lambda_1$ can be characterized as the best (the greatest) constant $C>0$ in the Poincar\'e inequality \begin{equation}\label{1.4} \int_{\Omega} |\nabla u|^p \geq C \int_{\Omega} |u|^p \end{equation} for all $u\in W^{1,p}_0 (\Omega)$, where identity $$ \int_{\Omega} |\nabla u|^p - \lambda_1 \int_{\Omega}|u|^p =0 $$ holds exactly for the multiples of the first eigenfunction $\varphi_1$. Let us recall (see e.g. [9, pp. 114, 115]) that, for every $h\in L^{\infty}(\Omega)$, the problem \begin{equation}\label{1.5} \begin{gathered} \Delta_p u = h \quad\mbox{in } \Omega, \\ u =0 \quad\mbox{on } \partial \Omega, \end{gathered} \end{equation} has a unique solution $u \in W^{1,p}_0 (\Omega) \cap C^{1,\nu} (\bar \Omega)$. Moreover, since $C^{1,\nu}(\bar \Omega)$ is compactly imbedded into $C^1(\bar \Omega)$, we can introduce the compact operator $$\Delta_p^{-1} \colon L^{\infty} (\Omega) \to C^1 (\bar \Omega) $$ such that $u= \Delta_p^{-1} h$ is the unique solution of (\ref{1.5}). In particular, every solution of (\ref{1.1}) belongs to $C^1_0 (\bar \Omega)$. In our further considerations we will use the standard spaces $W^{1,p}_0 (\Omega),\\ L^p (\Omega), C(\bar \Omega)$ and $C^1 (\bar \Omega)$ (or $C^1_0 (\bar\Omega)$, respectively), with corresponding norms \begin{gather*} \|u\|= \Big(\int_{\Omega} |\nabla u|^p\Big)^{1/p},\quad \|u\|_{L^p} = \Big(\int_{\Omega} |u|^p\Big)^{1/p}, \\ \|u\|_C= \max_{x\in \Omega}|u(x)|, \quad \|u\|_{C^1}= \|u\|_C+\max_{x\in \Omega}|\nabla u(x)|, \end{gather*} respectively, (here $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}$ or $\mathbb{R}^N$). The subscript $0$ indicates that the traces (or values) of functions are equal zero on $\partial \Omega$. Moreover, for the element $h$ of any of the above mentioned space we use the following ($L^2$--orthogonal) decomposition $$ h(x) = \tilde h (x) + \bar{h} \varphi_1 (x), $$ and also $L^2$--nonorthogonal decomposition $$ h(x) = \tilde h (x) + \hat h, $$ where $\bar{h}, \hat h \in \mathbb{R}$ and $$ \int_{\Omega} \tilde h (x) \varphi_1(x) dx =0. $$ The particular subspaces formed by $\tilde h (x) $ will be denoted by $\tilde W^{1,p}_0 (\Omega), \tilde C (\bar \Omega)$, and $\tilde C^1_0 (\bar\Omega)$, respectively. By $B_X (v,\rho)$ we denote the open ball in the space $X$ with the center $v$ and radius $\rho$, where $X = C (\bar \Omega)$ or $X= C^1_0 (\bar \Omega)$. We introduce the energy functional associated with (\ref{1.1}): $$ E_f (u)\colon = \frac{1}{p} \int_{\Omega} |\nabla u|^p - \frac{\lambda_1}{p} \int_{\Omega} |u|^p - \int_{\Omega} f u, \, \, u \in W^{1,p}_0 (\Omega). $$ This functional is continuously Fr\'echet differentiable on $W^{1,p}_0 (\Omega)$ and its {\it critical points} correspond one--to--one to {\it solutions} of (\ref{1.1}). Our main results concern the geometry of $E_f$ and the structure of the set of its critical points on one hand and the solvability properties of (\ref{1.1}) on the other hand. They are formulated in theorems below. \begin{theorem}\label{th1.1} Let $1 0$ such that for any $f\in B_C (\tilde f, \rho)$ the functional $E_f$ is unbounded from below and has at least one critical point. Moreover, for $f\in B_C (\tilde f, \rho) \setminus \tilde C (\bar \Omega)$ the functional $E_f$ has at least two distinct critical points. \end{theorem} \begin{theorem}\label{th1.2} Let $p>2$ and $0\neq \tilde f \in \tilde C (\bar\Omega)$. Then the functional $E_{\tilde f}$ is bounded from below and has at least one critical point (which is the global minimizer). Moreover, there exists $\rho = \rho(\tilde f) >0$ such that for $f\in B_C (\tilde f, \rho) \setminus \tilde C (\bar \Omega)$ the functional $E_f$ has at least two distinct critical points. \end{theorem} \begin{theorem}\label{th1.3} Let $p>1, p\neq 2, \tilde f \in \tilde C (\bar\Omega)$. Then the problem (\ref{1.1}) has at least one solution if $f=\tilde f$. For $0\neq \tilde f \in \tilde C (\bar \Omega)$ there exists $\rho = \rho (\tilde f) >0$ such that (\ref{1.1}) has at least one solution for any $f\in B_C (\tilde f, \rho)$. Moreover, there exist real numbers $F_- < 0 < F_+$ (see Fig. 1) such that the problem (\ref{1.1}) with $f= \tilde f + \hat{f}$ has \begin{itemize} \item[(i)] No solution for $\hat{f} \notin [F_-, F_+]$ \item[(ii)] At least two distinct solutions for $\hat{f} \in (F_-, 0) \cup (0, F_+)$ \item[(iii)] At least one solution for $\hat{f} \in \{F_-, 0, F_+\}$. \end{itemize} \end{theorem} \begin{figure}[t] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(60,60)(-30,-25) \put(-30,0){\line(1,0){60}} \put(0,-25){\line(0,1){50}} \dashline{2}(-16,16)(16,16) \put(-25,-25){\line(1,1){50}} \put(-25,25){\line(1,-1){50}} \put(29.8,-0.86){$\to$} \put(-0.87,25.7){$\uparrow$} \put(-16,15){\line(0,1){2}} \put(-16,-1){\line(0,1){2}} \put(-19,-4){$F_-$} \put(16,15){\line(0,1){2}} \put(16,-1){\line(0,1){2}} \put(14,-4){$F_+$} \put(28,-1){\line(0,1){2}} \put(24,-5){$\tilde f= 1$} \put(28,16){$C(\bar{\Omega})$} \put(2,18){$\tilde f$} \put(-9,28){$\tilde C(\bar{\Omega})$} \end{picture} \end{center} \caption{\label{fig:1} ``Slice'' of $C(\bar \Omega)$ containing all constants and one fixed $\tilde f \in \tilde C(\bar\Omega)$.} \end{figure} \begin{remark}\label{rem1.1} \rm Note that standard bootstrap regularity argument implies that any solution from Theorems \ref{th1.1}--\ref{th1.3} belongs to $L^{\infty}(\Omega)$ (cf. Dr\'abek, Kufner, Nicolosi \cite{10}). It follows then from the regularity results of Tolksdorf \cite{23} (see also Di Benedetto \cite{6} and Liebermann \cite{16}) that it belongs to $C^{1,\nu}(\bar\Omega)$ with some $\nu \in (0,1)$. In particular, our solution is an element of $C^1_0 (\bar\Omega)$. \end{remark} \begin{remark}\label{rem1.2} \rm In particular, it follows from our results that the set of $f\in C (\bar \Omega)$ for which (\ref{1.1}) has at least one solution has a nonempty interior in $C(\bar\Omega)$. \end{remark} \begin{remark}\label{rem1.3} \rm Note that Theorem \ref{th1.3} provides necessary and sufficient condition for solvability of the problem (\ref{1.1}). This condition is in fact of Landesman--Lazer type (see \cite{15}, cf. also \cite{11}). Indeed, given $\tilde f \in \tilde C (\bar \Omega), \tilde f \neq 0$, the problem (\ref{1.1}) with the right hand side $f(x) = \tilde f(x) + \hat{f}$ has a solution if and only if $$ F_-(\tilde f) \leq \frac{1}{\|\varphi_1\|_{L^1}} \int_{\Omega} f(x) \varphi_1 (x) dx \leq F_+ (\tilde f). $$ However, it should be pointed out that this condition differs from the original condition of Landesman and Lazer due to the fact that $F_-$ and $F_+$ depend on the component $\tilde f$ of the right hand side $f$ and not on the perturbation term (which is actually not present in our problem (\ref{1.1})). By homogeneity we have that for any $t>0$, $$ F_{\pm} (t \tilde f) = t F_{\pm} (\tilde f). $$ \end{remark} Our proofs rely on the combination of the variational approach and the method of lower and upper solutions. We also use essentially the results obtained by Dr\'abek and Holubov\'a \cite{8}, Tak\'a\v c \cite{21} and Fleckinger--Pell\'e and Tak\'a\v c \cite{14}. In fact, Theorem \ref{th1.1} was proved already in \cite{8}, however, here different approach is used. During the preparation of this manuscript the author received preprint of Tak\'a\v c \cite{22}, where similar result to our Theorem \ref{th1.3} is proved. However, the approach used in \cite{22} is very different from ours. Our objective in this paper is to {\it avoid} complicated {\it technical assumptions}. For this reason we restrict to rather special domains $\Omega$ and right hand sides $f$. On the other hand, we belive that in our approach the main ideas appear more clearly and that possible generalization of $\Omega$ or $f$ will not bring any new insight neither into the geometry of $E_f$ nor to the solvability of (\ref{1.1}). It should be mentioned that our approach covers also the case $N=1$, and completes thus previous results in this direction proved by Del Pino, Dr\'abek and Man\'asevich \cite{5}, Dr\'abek, Girg and Man\'asevich \cite{7}, Man\'asevich and Tak\'a\v c \cite{18}, Binding, Dr\'abek and Huang \cite{3}, Dr\'abek and Tak\'a\v c \cite{12}. In fact, the first relevant result which led to better understanding of the problem appeared in \cite{5}. Note also that our Theorems \ref{th1.1}, \ref{th1.2} and \ref{th1.3} express not only the difference between the linear case $p=2$ and the nonlinear case $p\neq 2$ but also the striking difference between the case $12$. The main goal of this paper is actually to emphasize this fact. \section {Auxiliary assertions, survey of known facts} It should be pointed out that $E_f$ is continuously differentiable and weakly lower semicontinuous functional on $W^{1,p}_0 (\Omega)$. The following notions are crutial in the study of the geometry of the functional $E_f$. \begin{definition}\label{def2.1} \rm We say that the functional $$ E_f \colon W^{1,p}_0 (\Omega) \to \mathbb{R} $$ has a {\it local saddle point geometry} if we can find $u, v \in W^{1,p}_0 (\Omega)$ which are separated by $\tilde W^{1,p}_0 (\Omega)$ in the sense that $$ E_f(u) < \inf_{w\in \tilde W^{1,p}_0 (\Omega)} E_f (w), \quad E_f (v) < \inf_{w\in \tilde W^{1,p}_0 (\Omega)} E_f (w) $$ and any continuous path from $u$ to $v$ in $W^{1,p}_0 (\Omega)$ has a nonempty intersection with $\tilde W^{1,p}_0 (\Omega)$. We say that $E_f$ has a {\it local minimizer geometry} if we can find open bounded set $D\subset W^{1,p}_0 (\Omega)$ such that $$ \inf_{u\in D} E_f (u) < \inf_{u \in \partial D} E_f (u). $$ The following lemma is crutial for application of variational methods. Its proof can be found in [8, Lemma 2.2] (or in [7, Proposition 2.1] in one dimensional case). \end{definition} \begin{lemma} \label{lem2.1} Let $p>1$, $f = \tilde f + \bar f \varphi_1$ with $\bar f \neq 0$. Then $E_f$ satisfies Palais--Smale (PS) condition, i.e. if $E_f (u_n) \to c \in \mathbb{R}, \, \, E'_f (u_n) \to 0$ then $\{u_n\}$ contains strongly convergent subsequence in $W^{1,p}_0 (\Omega)$. \end{lemma} Note that the assertion of Lemma \ref{lem2.1} is not true if $\bar f =0 $ (see \cite{5}). The following assertion deals with the case $12$ and prove that $E_f$ has a local minimizer geometry in this case. The following notions are crutial in the application of the method of lower and upper solutions. \begin{definition}\label{def2.2} \rm A function $u_s \in C^1 (\bar \Omega)$ is an {\it upper solution} of (\ref{1.1}) if \begin{gather*} \int_{\Omega} |\nabla u_s|^{p-2} \nabla u_s \cdot \nabla v - \lambda_1 \int_{\Omega} |u_s|^{p-2} u_s v \geq \int_{\Omega} f v \quad \forall v \in W^{1,p}_0 (\Omega), v \geq 0,\\ u_s \geq 0 \quad \mbox{on } \partial \Omega. \end{gather*} \end{definition} In an analogous way we define {\it a lower solution} $u_l$ of (\ref{1.1}). \begin{definition}\label{def2.3} \rm Let $u, v \in C^1 (\bar \Omega)$. We say that $u \prec v$ if $u(x) < v(x)$ on $\Omega$, and for $x\in \partial \Omega$ either $u(x) < v(x)$, or $u(x)=v(x)$ and $(\partial u/\partial n) (x)> (\partial v/\partial n) (x)$. \end{definition} \begin{definition}\label{def2.4} \rm A {\it lower solution} $u_l$ of (\ref{1.1}) is said to be {\it strict} if every solution $u$ of (\ref{1.1}) such that $u_l \leq u$ on $\Omega$ satisfies $u_l \prec u$. In an analogous way we define a {\it strict upper solution} of (\ref{1.1}). \end{definition} For $h \in C (\bar \Omega)$ we define an operator $T_f \colon C^1_0 (\bar\Omega) \to C^1_0 (\bar\Omega)$ as $T_f (v) = u$ where $u$ satisfies \begin{gather*} \Delta_p u = f(x) -\lambda_1|v|^{p-2} v \quad\mbox{in } \Omega, \\ u=0 \quad\mbox{on } \partial \Omega. \end{gather*} The operator $T_f$ is compact and its fixed points, i.e. $u = T_f (u)$ $u\in C^1_0(\Omega)$, correspond to solutions of the original problem (\ref{1.1}). The following assertions are proved in \cite{8}, the idea comes from \cite{4}. \begin{lemma}[Well--Ordered Lower and Upper Solutions] \label{lem2.3} Let $u_l$ and $u_s$ be lower and upper solutions, respectively, of (\ref{1.1}) such that $u_l \leq u_s$. Then the problem (\ref{1.1}) has at least one solution $u$ satisfying $$ u_l \leq u \leq u_s \quad \mbox{ in } \Omega. $$ If, moreover, $u_l$ and $u_s$ are strict and satisfy $u_l \prec u_s$, then there exists $R_0 >0$ such that for all $R\geq R_0$ $$ \deg[I -T_f; \mathcal M_1, 0]=1, $$ where $\mathcal M_1 = \{u\in C^1_0 (\bar\Omega); u_l\prec u \prec u_s\} \cap B_{C_0^1} (0,R)$. \end{lemma} \begin{lemma}[Non--Ordered Lower and Upper Solutions]\label{lem2.4} Let $u_l$ and $u_s$ be lower and upper solutions, respectively, of (\ref{1.1}) and $ u_l (x_0) > u_s (x_0)$ for some $x_0 \in \Omega$. Then (\ref{1.1}) has at least one solution in the closure (with respect to $C^1$-norm) of the set $$ S=\{u \in C^1_0 (\bar\Omega); x_1, x_2 \in \Omega \colon \, \, u(x_1) u_s (x_2)\}. $$ Set $\mathcal M_2 = S\cap B_{C^1_0} (0, R)$ and assume that there is no solution of (\ref{1.1}) on $\partial \mathcal M_2$. Then there exists $R_0> 0$ such that for all $R\geq R_0$ $$ \deg [I- T_f; \mathcal M_2, 0]= -1. $$ \end{lemma} As an immediate consequence of Lemmas \ref{lem2.3} and \ref{lem2.4} we have the following proposition. \begin{proposition}\label{prop2.1} Let (\ref{1.1}) be solvable for $f_1\in C(\bar \Omega)$ and $f_2\in C (\bar\Omega)$ such that $f_1(x) \leq f_2 (x)$, $x \in \bar \Omega$. Then it is also solvable for any $f \in C(\bar \Omega)$ such that $f_1 (x) \leq f(x)\leq f_2 (x), x\in \bar\Omega$. \end{proposition} \paragraph{Proof.} Let $u_i$ be a solution of (\ref{1.1}) with $f_i, i=1,2$. Then $u_l = u_1$ and $u_s = u_2$ are lower and upper solutions, respectively, of (\ref{1.1}) with $f$. Then either Lemma \ref{lem2.3} or \ref{lem2.4} applies to get a solution. \hfill$\square$ The following assertion deals with the case $p>2$ and helps to get the information about the geometry of the energy functional $E_f$. \begin{proposition}[{[14, Theorem 1.1]}]\label{prop2.2} There exists a positive constant $C= C(p,\Omega)$ such that for all $u\in W^{1,p}_0 (\Omega), u(x) = \tilde u(x) +\bar u \varphi_1(x)$, $$ \int_{\Omega} |\nabla u|^p -\lambda_1 \int_{\Omega} |u|^p \geq C\Big(|\bar u|^{p-2} \int_{\Omega} |\nabla \varphi_1|^{p-2} |\nabla \tilde u|^2 + \int_{\Omega} |\nabla \tilde u|^p\Big). $$ \end{proposition} We will need also the following imbedding type inequality (see [21, Lemma 4.2], [14, Lemma 4.2]): Let $p>2$, then there exists $\tilde C >0$ such that for all $u\in W^{1,p}_0 (\Omega)$, \begin{equation}\label{2.1} \Big(\int_{\Omega} |u|^2\Big)^{1/2} \leq \tilde C \Big(\int_{\Omega} |\nabla \varphi_1|^{p-2} |\nabla u|^2\Big)^{1/2}. \end{equation} The last assertion of this section is related to the application of the degree argument in the proof of Theorem \ref{th1.3}. \begin{proposition}[see {[21, Theorems 2.3 and 2.8]}] \label{prop2.3} Let $p>1$ and $K$ be a compact set in $C(\bar \Omega)$ and $\int_{\Omega} f \varphi_1 \neq 0$ for any $f\in K$. Then there exists a constant $\tilde C_1= \tilde C_1 (K)>0$ such that $$ \|u\|_{C^1_0} \leq \tilde C_1 $$ for any possible solution $u$ of (\ref{1.1}) with $f\in K$. \end{proposition} \section{Proof of Theorem \ref{th1.1}} For the case $10$ independent of $n$. Dividing the last inequality by $\|\tilde u_n\|^p$ and passing to the limit for $n\to \infty$, we obtain $$ \lim_{n\to \infty}\{\frac{1}{p} \int_{\Omega} |\frac{\bar u\nabla \varphi_1}{\|\tilde u_n\|}+ \nabla \hat{\tilde u}_n|^p - \frac{\lambda_1}{p} \int_{\Omega} |\frac{\bar u\varphi_1}{\|\tilde u_n\|}+ \hat {\tilde u}_n|^p -\int_{\Omega} \tilde f \frac{\tilde u_n}{\|\tilde u_n\|^p} \} \leq 0, $$ where $\hat {\tilde u}_n = \frac{\tilde u_n}{\|\tilde u_n\|}$. The closedness of $\tilde W^{1,p}_0 (\Omega)$ and the compactness of the imbedding $W^{1,p}_0 (\Omega) \hookrightarrow \hookrightarrow L^p (\Omega)$ imply that there exists $\tilde u_0 \in \tilde W^{1,p}_0 (\Omega), \|\tilde u_0\|= 1$, such that $$ \frac{1}{p} \int_{\Omega} |\nabla \tilde u_0|^p - \frac{\lambda_1}{p} \int_{\Omega} |\tilde u_0|^p=0. $$ However, this contradicts the variational characterization and the simplicity of $\lambda_1$. \begin{lemma}\label{lem3.1} Let $\tilde u_{\bar u} \in \tilde W^{1,p}_0 (\Omega)$ be as above. Then $\|\tilde u_{\bar u}\|_{L^p} = o(\bar u)$ as $|\bar u|\to \infty$. \end{lemma} \paragraph{Proof.} (i) Assume that there exists $\{\bar u_n\} \subset \mathbb{R}$ such that $\bar u_n\to \infty$ and \begin{equation}\label{3.2} \frac{\bar u_n}{\|\tilde u_{\bar u_n}\|} \to 0. \end{equation} Set $\hat{\tilde u}_{\bar u_n} = \tilde u_{\bar u_n}/\|\tilde u_{\bar u_n}\|$. It follows from (\ref{3.1}) that \begin{multline}\label{3.3} \liminf_{\bar u_n\to \infty} \Big\{\frac{1}{p} \int_{\Omega} |\frac{\bar u_n}{\|\tilde u_{\bar u_n\|}} \nabla \varphi_1 + \nabla \hat {\tilde u}_{\bar u_n}|^p - \frac{\lambda_1}{p} \int_{\Omega} |\frac{\bar u_n}{\|\tilde u_{\bar u_n}\|} \varphi_1 + \hat {\tilde u}_{\bar u_n}|^p \\ -\frac{1}{\|\tilde u_{\bar u_n}\|^{p-1}} \int_{\Omega} \tilde f \hat {\tilde u}_{\bar u_n}\Big\} \leq 0. \end{multline} Passing to a subsequence if necessary we conclude $\hat {\tilde u}_{\bar u_n} \rightharpoonup u_0 \mbox{ in } W^{1,p}_0 (\Omega), \hat {\tilde u}_{\bar u_n} \to u_0 \mbox{ in } L^p (\Omega) \mbox{ and }$ \begin{equation}\label{3.4} \int_{\Omega} u_0 \varphi_1 = 0. \end{equation} At the same time, for large $u\in \mathbb{N}$, we have $$ \frac{1}{p}\int_{\Omega} |\frac{\bar u_n}{\|\tilde u_{\bar u_n}\|} \nabla \varphi_1 + \nabla \hat {\tilde u}_{\bar u_n}|^p \geq \varepsilon $$ with some $\varepsilon >0$. It follows then from (\ref{3.3}) that $$ \frac{\lambda_1}{p} \int_{\Omega} |u_0|^p \geq \varepsilon $$ which means that $u_0 \neq 0$. At the same time we get from (\ref{3.3}) that $$ \frac{1}{p} \int_{\Omega} |\nabla u_0|^p - \frac{\lambda_1}{p} \int_{\Omega} |u_0|^p \leq 0 $$ and so the variational characterization and simplicity of $\lambda_1$ imply that $u_0 = k\varphi_1, k\neq 0$. But this contradicts (\ref{3.4}). (ii) Assume that $\bar u_n \to \infty$ and there exist constant $C>0$ independent of $n$ such that \begin{equation}\label{3.5} \frac{\|\tilde u_{\bar u_n}\|}{\bar u_n} \leq C. \end{equation} It follows from (\ref{3.1}) that \begin{equation}\label{3.6} \lim_{\bar u_n\to \infty} \inf \{\frac{1}{p} \int_{\Omega} |\nabla \varphi_1+\nabla (\frac{\tilde u_{\bar u_n}}{\bar u_n})|^p -\frac{\lambda_1}{p} \int_{\Omega} |\varphi_1+\frac{\tilde u_{\bar u_n}}{\bar u_n}|^p -\int_{\Omega} \tilde f \frac{\tilde u_{\bar u_n}}{\bar u_n^p}\} \leq 0. \end{equation} Passing to a subsequence if necessary, we conclude that there is $u_0 \in W^{1,p}_0 (\Omega)$ such that $\frac{\tilde u_{\bar u_n}}{\bar u_n} \rightharpoonup u_0$ in $W^{1,p}_0 (\Omega), \frac{\tilde u_{\bar u_n}}{\bar u_n} \to u_0$ in $L^p (\Omega)$ and $$ \int_{\Omega} u_0 \varphi_1 =0. $$ Let $u_0 \neq 0$. Then we get from (\ref{3.6}) that $$ \frac{1}{p} \int_{\Omega} |\nabla \varphi_1 + \nabla u_0|^p - \frac{\lambda_1}{p} \int_{\Omega} |\varphi_1 + u_0|^p \leq 0, $$ which contradicts the variational characterization and simplicity of $\lambda_1$. Hence $u_0 = 0$, i.e. \begin{equation}\label{3.7} \frac{\tilde u_{\bar u_n}}{\bar u_n} \to 0 \quad \mbox{in } L^p(\Omega). \end{equation} Assume now that the assertion of lemma is not true. Then there is a sequence $\{\bar u_n\} \subset \mathbb{R}$, $\bar u_n \to\infty$, such that for some $\tilde C_2>0$ we have $$ \frac{\|\tilde u_{\bar u_n}\|_{L^p}}{\bar u_n} \geq \tilde C_2. $$ For such a sequence we have that either (\ref{3.2}) or (\ref{3.5}) holds. The former case is impossible by (i) the latter case contradicts (\ref{3.7}). \hfill$\square$ As a consequence of Lemma \ref{lem3.1} we have \begin{equation}\label{3.8} \min_{\tilde u\in \tilde W^{1,p}_0 (\Omega)} \{\frac{1}{p} \int_{\Omega} |\bar u\nabla \varphi_1 + \nabla \tilde u|^p -\frac{\lambda_1}{p} \int_{\Omega} |\bar u\varphi_1+\tilde u|^p - \int_{\Omega} \tilde f \tilde u\} = o (\bar u),\; |\bar u|\to \infty. \end{equation} \begin{lemma}\label{lem3.2} For a given $T>0$ there exists $R>0$ such that for any $\bar u\in [0,T]$ and $\tilde u\in \tilde W^{1,p}_0 (\Omega), \|\tilde u \|=R$, we have \begin{equation}\label{3.9} \frac{1}{p} \int_{\Omega} |\bar u\nabla \varphi_1+\nabla \tilde u|^p -\frac{\lambda_1}{p} \int_{\Omega}|\bar u \varphi_1+\tilde u|^p -\int_{\Omega} \tilde f \tilde u \geq 0. \end{equation} \end{lemma} \paragraph{Proof.} Assume that there is $T>0, \bar u_n\in [0,T], \|\tilde u_n\|\to \infty$ such that \begin{equation}\label{3.10} \frac{1}{p} \int_{\Omega} |\bar u_n \nabla \varphi_1+ \nabla \tilde u_n|^p - \frac{\lambda_1}{p} \int_{\Omega} |\bar u_n \varphi_1 + \tilde u_n|^p-\int_{\Omega} \tilde f \tilde u_n <0. \end{equation} Set $\hat {\tilde u}_n = \tilde u_n/\|\tilde u_n\|$. Passing to subsequences if necessary we can assume that $\hat {\tilde u} \rightharpoonup u_0$ in $W^{1,p}_0 (\Omega), \int_{\Omega} u_0 \varphi_1 =0$, $\bar u_n \to \bar u_0 \in [0,T]$. At the same time, dividing (\ref{3.10}) by $\|\tilde u_n\|^p$, passing to the limit for $n\to\infty$ we derive that $u_0 \neq 0$ and $$ \frac{1}{p} \int_{\Omega} |\nabla u_0|^p - \frac{\lambda_1}{p}\int_{\Omega} |u_0|^p \leq 0 $$ which contradicts the variational characterization and simplicity of $\lambda_1$. \hfill$\square$ Let $\rho>0$ be small enough (to be specified later) and consider $f\in B_C(\tilde f,\rho) \setminus \tilde C(\bar \Omega)$. Then $f$ splits as follows: $$ f(x) = \tilde f (x) + \bar f \varphi_1 (x) $$ with $|\bar f|$ small, $\bar f \neq 0$. Then \begin{align*} E_f(u) =&\frac{1}{p} \int_{\Omega} |\nabla u|^p - \frac{\lambda_1}{p} \int_{\Omega} |u|^p - \int_{\Omega} \tilde f \tilde u - \bar u \int_{\Omega} \bar f \varphi_1 \\ =& E_{\tilde f} (u) - \bar u \int_{\Omega} \bar f \varphi_1, \quad u\in W^{1,p}_0 (\Omega), \end{align*} where $u= \bar u\varphi_1 +\tilde u$. Let $\bar f<0$, so $\bar f \in (-\bar \rho, 0)$ with small $\bar\rho >0$. We shall construct the set $$ D= \{u\in W^{1,p}_0 (\Omega)\colon u = \bar u\varphi_1 +\tilde u, \bar u \in (0,T), \|\tilde u\|0$ and $R>0$ to be specified later. We choose $T_1 >0$ so that \begin{equation}\label{3.11} E_{\tilde f} (\tilde u_{T_1}) \leq 2 E_{\tilde f} (\tilde u_0) \end{equation} (this is possible due to (\ref{3.1}), remind that $\tilde u_{T_1}$ and $\tilde u_0$ are the points where $\inf_{\tilde u \in \tilde W^{1,p}_0(\Omega)} E_{\tilde f} (\bar u \varphi_1 +\tilde u)$ is achieved for $\bar u =T_1$ and $\bar u=0$, respectively). Then take $\rho >0$ (and hence $\bar \rho >0$) so small that \begin{equation}\label{3.12} E_f (T_1 \varphi_1 + \tilde u_{T_1}) \leq \frac{3}{2} E_{\tilde f} (\tilde u_0) \end{equation} if $f\in B_C (\tilde f, \rho) \setminus \tilde C(\bar \Omega)$. Now we choose $T>0$ so that \begin{equation}\label{3.13} E_f (T \varphi_1+ \tilde u_T) \geq 0 \end{equation} (this is possible due to (\ref{3.8}) and $\bar f<0$). Finally, we choose $R=R(T)>0$ according to Lemma \ref{lem3.2} (see Fig. 2). Then it follows from Lemma \ref{lem3.2}, (\ref{3.12}) and (\ref{3.13}) that \begin{equation}\label{3.14} \inf_{u\in D} E_f (u) < \inf_{u\in \partial D} E_f (u). \end{equation} \begin{figure}[t] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(65,30)(-15,-15) \put(-5,0){\line(1,0){50}} \put(0,-15){\line(0,1){30}} \put(0,-10){\line(1,0){35}} \put(0,10){\line(1,0){35}} \put(35,10){\line(0,-1){20}} \put(-0.87,15.5){$\uparrow$} \put(44.8,-0.86){$\to$} \put(-7,-1){$\left\{\vbox to 11.2mm{}\right.$} \put(-21,-1){$\|\tilde u\|\leq R$} %\put(-4,1){$R$} \put(14,-1){\line(0,1){2}} \put(12,-4){$\varphi_1$} \put(36,-4){$T\varphi_1$} \put(18,4){$D$} \put(-15,15){$\widetilde W_0^{1,p}(\Omega)$} \put(38,12){$W_0^{1,p}(\Omega)$} \end{picture} \end{center} \caption{\label{fig:2} The set $D$ constructed in the proof of Theorem \ref{th1.1}} \end{figure} Since $E_f$ is weakly lower semicontinuous functional on $D$ there exists a global minimizer of $E_f$ in $D$. Let $u_D \in D$ be the point of global minimum, i.e. $$ E_f (u_D) = \min_{u\in D} E_f(u). $$ Note that $E_f$ is unbounded from below. This is easy to see, choosing e.g. $u_n= \bar u_n\varphi_1, \bar u_n \to -\infty$, we obtain $E_f (u_n) \to -\infty$. So, $E_f$ has a Mountain Pass Theorem Geometry. Because $E_f$ satisfies also (PS) condition according to Lemma \ref{lem2.1}, we can apply the results of Rabinowitz \cite{20} to derive the existence of $u_0 \in W^{1,p}_0 (\Omega), u_0 \neq u_D$, which is also a critical point of $E_f$. To summarize, we proved that for $f\in B_C (\tilde f, \rho)\setminus \tilde C(\bar \Omega)$ the functional $E_f$ has at least two distinct critical points. The case $\bar f>0$ is similar. It remains to prove that $E_{\tilde f}$ has at least one critical point. This follows from the argument based on the method of upper and lower solutions. It follows from the previous considerations that there is $\bar f >0$ small enough such that $E_{\tilde f\pm\bar f\varphi_1}$ has critical points $u_{\pm} \in W^{1,p}_0 (\Omega)$, i.e. $$ \int_{\Omega} |\nabla u_{\pm}|^{p-2} \nabla u_{\pm} \cdot \nabla v - \lambda_1 \int_{\Omega} |u_{\pm}|^{p-2} u_{\pm} v = \int_{\Omega} \tilde f v \pm \int_{\Omega} \bar f \varphi_1 v $$ holds for any $v\in W^{1,p}_0 (\Omega)$. It follows from Proposition \ref{prop2.1} that there is a solution $u\in W^{1,p}_0 (\Omega)$ satisfying $$ \int_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v - \lambda_1 \int_{\Omega} |u|^{p-2} u v = \int_{\Omega} \tilde f v $$ for any $v\in W^{1,p}_0 (\Omega)$. This is equivalent to the fact that $u$ is a critical point of $E_{\tilde f}$. This completes the proof of Theorem \ref{th1.1}. \section{Proof of Theorem \ref{th1.2}} We consider the case $p>2$ and the energy functional $E_{\tilde f}$ with $\tilde f\in \tilde C (\bar \Omega), \tilde f \neq 0$. Let us choose a function $\varphi \in W^{1,p}_0 (\Omega), \varphi \geq 0$ in $\Omega$ and such that $$ \{x\in \Omega \colon \varphi (x) >0\} \subset \{ x \in \Omega \colon \tilde f (x) >0\} $$ (note that this is possible because the latter set is an open subset of $\Omega$). Then there exists $t>0$ (small enough) such that for $v= t \varphi$ we have \begin{equation}\label{4.1} E_{\tilde f} (v) <0. \end{equation} Making use of Proposition \ref{prop2.2} the H\"older and Young inequalities we have the following estimate \begin{align*} E_{\tilde f} (u) \geq& \frac{C}{p} \Big[|\bar u|^{p-2} \int_{\Omega} |\nabla \varphi_1|^{p-2} |\nabla \tilde u|^2 + \int_{\Omega} |\nabla \tilde u|^p \Big]-\Big(\int_{\Omega} |\tilde f|^{p'}\Big)^{1/p'} \Big(\int_{\Omega} |\tilde u|^p\Big)^{1/p} \\ \geq & \frac{C}{p} \Big[|\bar u|^{p-2} \int_{\Omega} |\nabla \varphi_1|^{p-2} |\nabla \tilde u|^2+ \int_{\Omega} |\nabla \tilde u|^p\Big]- \frac{C^p_1 \varepsilon^p}{p} \|\tilde u\|^p - \frac{1}{\varepsilon^p p'} \|\tilde f\|^{p'}_{L^{p'}}, \end{align*} where $C_1 >0$ is the constant of the imbedding $W^{1,p}_0 (\Omega) \hookrightarrow L^p (\Omega)$. Choosing $C^p_1 \varepsilon^p = \frac{C}{2}$ we arrive at \begin{equation}\label{4.2} E_{\tilde f} (u) \geq \frac{C}{2p} \|\tilde u\|^p + \frac{C}{p} |\bar u|^{p-2} \int_{\Omega} |\nabla \varphi_1|^{p-2} |\nabla \tilde u|^2 - \frac{(\frac{2}{C})^{\frac{1}{p-1}} C_1^{1-\frac{1}{p}}}{p'} \|\tilde f\|^{p'}_{L^{p'}}. \end{equation} It follows from here that there exists $R=R(\tilde f)>0$ such that for any $u= \bar u \varphi_1+\tilde u \in W^{1,p}_0 (\Omega) $ with $\|\tilde u\|= R$ we have \begin{equation}\label{4.3} E_{\tilde f} (u) >0. \end{equation} Let us consider now $u= \bar u \varphi_1 + \tilde u \in W^{1,p}_0 (\Omega)$ for which \begin{equation}\label{4.4} \frac{C}{p} |\bar u|^{p-2} \int_{\Omega} |\nabla \varphi_1|^{p-2} |\nabla \tilde u|^2 \leq C_2 \|\tilde f\|^{p'}_{L^{p'}} \end{equation} where we denoted $C_2 = \frac{1}{p'} (\frac{2}{C})^{\frac{1}{p-1}} C_1^{1-\frac{1}{p}}$. It follows then from the H\"older inequality that $E_{\tilde f}(u) \geq - \|\tilde f\|_{L^2} \|\tilde u\|_{L^2}$. If we combine this with (\ref{2.1}) and (\ref{4.4}) we get \begin{equation}\label{4.5} E_{\tilde f} (u) \geq -\frac{\tilde C p^{1/2} C_2^{1/2} \|\tilde f\|_{L^2} \|f\|^{\frac{p'}{2}}_{L^{p'}}}{C^{1/2} |\bar u|^{\frac{p-2}{2}}}. \end{equation} Let us define the set $$ D= \{u\in W^{1,p}_0 (\Omega) \colon u = \bar u \varphi_1 + \tilde u, \quad \bar u \in (-T, T), \|\tilde u\|0$ to be fixed later (see Fig. 3). It follows from (\ref{4.1}) that $$ i\colon = \inf_{u\in D} E_{\tilde f}(u) <0 $$ independently of $T\gg 1$. It follows from (\ref{4.5}) that for $u= \pm T \varphi_1+\tilde u$ satisfying (\ref{4.4}) we have \begin{equation}\label{4.6} E_{\tilde f}(u) > i \end{equation} if $T$ is large enough. On the other hand we have directly from (\ref{4.2}) that \begin{equation}\label{4.7} E_{\tilde f}(u) \geq 0>i \end{equation} for $u= \pm T \varphi_1 + \tilde u$ which do not satisfy (\ref{4.4}). Now, if we combine (\ref{4.3}), (\ref{4.6}) and (\ref{4.7}), we get \begin{equation}\label{4.8} i< \inf_{u\in \partial D} E_{\tilde f}(u). \end{equation} Thus $E_{\tilde f}$ has a local minimizer geometry. In particular, it follows also from above considerations that $E_{\tilde f}$ is bounded from below on $W^{1,p}_0(\Omega)$. Since $E_{\tilde f}$ is weakly lower semicontinuous functional on the bounded, convex and closed set $\bar D$, it has to achieve its minimum there. Due to (\ref{4.8}) the minimizer is an interior point of $D$ and due to the differentiability of $E_{\tilde f}$ it is a critical point at the same time. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[t] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(100,30)(-53,-15) \put(-40,0){\line(1,0){85}} \put(0,-15){\line(0,1){30}} \put(-35,-10){\line(1,0){70}} \put(-35,10){\line(1,0){70}} \put(35,10){\line(0,-1){20}} \put(-35,10){\line(0,-1){20}} \put(-0.86,15.5){$\uparrow$} \put(44.8,-0.86){$\to$} \put(-43,-1){$\left\{\vbox to 11.2mm{}\right.$} \put(-57,-1){$\|\tilde u\|\leq R$} %\put(-4,1){$R$} \put(14,-1){\line(0,1){2}} \put(12,-4){$\varphi_1$} \put(36,-4){$T\varphi_1$} \put(-34,-4){$-T\varphi_1$} \put(18,4){$D$} \put(-15,15){$\widetilde W_0^{1,p}(\Omega)$} \put(38,12){$W_0^{1,p}(\Omega)$} \end{picture} \end{center} \caption{\label{fig:3} The set $D$ constructed in the proof of Theorem \ref{th1.2}} \end{figure} Let $\rho >0$ and consider $f\in B_C (\tilde f, \rho)\setminus \tilde C(\bar \Omega)$. Then, as in Section 3, split $f$ as follows: $$ f(x) = \tilde f(x) + \bar f \varphi_1 (x) $$ with $\bar f \neq 0$. Then $$ E_f (u) = E_{\tilde f} (u) - \bar u \int_{\Omega} \bar f \varphi_1 $$ and thus $E_f$ is unbounded from below (we can use the same reasoning as in the previous section). If $\rho$ is small enough (and so is $|\bar f|$) then inequality (\ref{4.8}) still holds. This means that $E_f$ has a Mountain Pass Theorem Geometry and we proceed exactly as in the previous section to conclude the existence of at least two distinct critical points of $E_{\tilde f}$. This completes the proof of Theorem \ref{th1.2}. \section{Proof of Theorem \ref{th1.3}} Let $\tilde f \in \tilde C (\bar\Omega)$. Then it follows from Theorems \ref{th1.1} and \ref{th1.3} that the problem (\ref{1.1}) has at least one weak solution. It follows from these theorems that for $\tilde f\neq 0$ there exists $\rho = \rho (\tilde f)>0$ such that (\ref{1.1}) has at least one solution for any $f\in B_C (\tilde f, \rho)$. So we shall concentrate to the proof of the second part of Theorem \ref{th1.3}. To this end we shall split $f\in C(\bar\Omega)$ as follows \begin{equation}\label{5.1} f(x) = \tilde f (x) + \hat{f}. \end{equation} Define $$ F_- = F_- (\tilde f):= \inf \hat{f}, \quad F_+ = F_+ (\tilde f) := \sup \hat{f}, $$ where the infimum and the supremum are taken over all $\hat{f}$ for which (\ref{1.1}) (with $f(x)$ given above) has a solution. It follows directly from the first part of Theorem \ref{1.3} that $F_-<0 F_-, \hat{f}_n \to F_-$ and denote by $u_n \in C^1_0 (\bar\Omega)$ corresponding solutions of (\ref{1.1}) with $f(x) = \tilde f(x) + \hat{f}_n$. According to Proposition \ref{prop2.3} the sequence $\{u_n\}$ is bounded in $C^1_0 (\bar\Omega)$. Compactness of $\Delta^{-1}_p$ implies the existence of a subsequence (denoted again by $\{u_n\}$) for which $u_n \to u_-$ in $C^1_0 (\bar\Omega)$ for some $u_- \in C^1_0 (\bar\Omega)$. Moreover, similarly as above, $u_-$ satisfies \begin{gather*} -\Delta_p u_- - \lambda_1 |u_-|^{p-2} u_- = \tilde f (x) + F_- \quad\mbox{in }\Omega \\ u_-= 0 \quad\mbox{on }\partial \Omega \end{gather*} Similarly, we prove that (\ref{1.1}) is solvable for $f(x)= \tilde f(x)+F_+$. This proves (iii). It remains to prove the multiplicity result stated in (ii). We proceed via contradiction. To this end we apply the degree theory combined with Lemmas \ref{lem2.3}, \ref{lem2.4} and Propositions \ref{prop2.1} and \ref{prop2.3}. Let us assume that $\hat{f} \in (0, F_+)$ (the proof in case $\hat{f} \in (F_-, 0)$ is similar). Then the problem (\ref{1.1}) with $f(x) = \tilde f(x) + \hat{f}$ has a solution $u$ and there exist $0<\hat{f}_1<\hat{f}<\hat{f}_2 < F_+$ such that (\ref{1.1}) has also solutions $u_i$ for $f_i(x) = \tilde f(x) + \hat{f}_i$, $i= 1,2$. It is straightforward to verify that $u_1$ and $u_2$ are lower and upper solutions, respectively, of (\ref{1.1}) with the right hand side $f$. We assume that $u$ is unique solution of (\ref{1.1}) obtained by Proposition \ref{prop2.1}, i.e. it is either $u_1\leq u \leq u_2 $ in $\Omega$ or $u\in \bar S$ (with $S$ defined in Lemma \ref{lem2.4}). Assume that the former case occurs, $u_1, u_2$ are strict, and $u_1\prec u_2$, i.e. $u\notin \partial \mathcal M_1$ with $R=R_0$ large enough (with $\mathcal M_1$ defined in Lemma \ref{lem2.3}). Then according to Lemma \ref{lem2.3}, we have that \begin{equation}\label{5.2} \deg [I-T_f; \mathcal M_1, 0] =1. \end{equation} Let us choose $\hat{f}_3 >F_+$. It follows from above considerations that (\ref{1.1}) with $f_3(x) = \tilde f(x)+ \hat{f}_3$ has no solution. Hence \begin{equation}\label{5.3} \deg [I -T_{f_3}; B_{C^1_0} (0,R), 0] =0 \end{equation} for arbitrary $R>0$. Consider now the family of functions $$ f_t (x) = \tilde f(x) + t\hat{f}+ (1-t)\hat{f}_3,\quad t\in [0,1]. $$ Then $K= \{f_t \in C (\bar\Omega)\colon t\in [0,1]\}$ is a compact subset of $C(\bar\Omega)$ and $$ H(t, \cdot) = I- T_{f_t},\quad t\in [0,1], $$ is a homotopy of compact perturbations of the identity. It follows from Proposition \ref{prop2.3} that for $R= R_1>R_0$ large enough we have that $$ \deg [I-T_{f_t}; B_{C^1_0} (0, R_1), 0] $$ is constant for $t\in [0,1]$. Due to (\ref{5.3}) we have also \begin{equation}\label{5.4} \deg [I- T_f; B_{C^1_0} (0,R_1), 0] = 0. \end{equation} Additivity property of the degree and (\ref{5.2}), (\ref{5.4}) imply that there is $\check{u}$ in $B_{C^1_0} (0,R_1)\setminus \mathcal M_1$ which is a solution of (\ref{1.1}) and evidently $\check{u} \neq u$ which is a contradiction with uniqueness of $u$. The proof follows the same lines if $u\in \bar S$ and $u\notin \partial \mathcal M_2$ (with $\mathcal M_2$ defined in Lemma \ref{lem2.4}). The only difference consists in substituting (\ref{5.2}) by $$ \deg[I-T_f; \mathcal M_2, 0] = -1. $$ Assume, now, that unique solution $u$ is obtained by means of Lemma \ref{lem2.3} but $u\in \partial \mathcal M_1$. Since $R_0$ can be chosen large enough this means that $u_1 \not\prec u$ or $u\not\prec u_2$. Let us assume $u_1\not\prec u$ (the other case is similar). This means that either there exists $x_0 \in \Omega$ such that $u_1(x_0) = u(x_0)$ or there exists $\check{x}_0 \in \partial \Omega$ such that $\frac{\partial u_1}{\partial n}(\check{x}_0) = \frac{\partial u}{\partial n}(\check{x}_0)$. We choose $\delta >0$ small enough (to be specified later) and define $u^{\delta}_1 (x) = u_1 (x) - \delta, x \in \Omega$. Then $u^{\delta}_1 \in C^1(\bar \Omega)$ and $u^{\delta}_1 \prec u$. We prove that for $\delta$ small this new function $u^{\delta}_1 $ is lower solution of (\ref{1.1}). Indeed, since $u_1 \in C (\bar\Omega)$, there exists a constant $C=C(\|u_1\|_C)>0$ such that for any $x\in \bar\Omega$, $$ \left||u_1(x) - \delta|^{p-2} (u_1 (x)-\delta) - |u_1(x)|^{p-2} u_1 (x)\right| \leq |\delta|^{p-1}, $$ for $12$. In either case, there exists $\delta_0>0$ such that for all $0<\delta <\delta_0$ we have \begin{equation}\label{5.5} \int_{\Omega} \left||u^{\delta}_1 (x)|^{p-2} u_1^{\delta} (x) - |u_1(x)|^{p-2} u_1(x)\right|\psi (x) dx \leq \frac{\hat{f}- \hat{f}_1}{2\lambda_1} \int_{\Omega} \psi (x)dx \end{equation} for all $\psi \geq 0, \psi \in W^{1,p}_0 (\Omega).$\\ Since $\nabla u^{\delta}_1(x) = \nabla u_1(x), x\in \Omega$, it follows from (\ref{5.5}) that $$ \int_{\Omega} |\nabla u^{\delta}_1|^{p-2} \nabla u_1^{\delta} \cdot \nabla \psi - \lambda_1 \int_{\Omega} |u^{\delta}_1|^{p-2} u_1^{\delta} \psi \leq \int_{\Omega} \tilde f \psi + \bar f \int_{\Omega} \psi, $$ for any $\psi \geq 0, \psi \in W^{1,p}_0 (\Omega)$, i.e. $u^{\delta}_1$ is a lower solution of (\ref{1.1}). Similarly we can define an upper solution $u_2^{\delta} = u_2 +\delta$ such that $u\prec u^{\delta}_2$. We define then a new set $\mathcal M^{\delta}_1$ by means of $u_1^{\delta}, u_2^{\delta}$, with $u_1^{\delta} \prec u_2^{\delta}$, and since $u\notin \partial \mathcal M_1^{\delta}$, we proceed as above to get a contradiction with the uniqueness of $u$. Assume, now, that unique solution $u$ is obtained by means of Lemma \ref{lem2.4} but $u\in \partial \mathcal M_2$. Since $R_0$ can be chosen large enough this means that we have two similar possibilities (which can occur simultaneously):\\ (i) either $u(x)\geq u_1(x), x\in \Omega$, and there exists $x^l_0 \in \Omega$ such that $u(x^l_0) = u_1 (x^l_0)$ or there exists $\check{x}^l_0 \in \partial \Omega$ such that $\frac{\partial u_1}{\partial n}(\check{x}^l_0)= \frac{\partial u}{\partial n}(\check{x}^l_0)$,\\ (ii) either $u(x) \leq u_2(x), x\in \Omega$, and there exists $x^s_0 \in \Omega$ such that $u(x^s_0) =u_2 (x^s_0)$ or there exists $\check{x}^s_0 \in \partial \Omega$ such that $\frac{\partial u_2}{\partial n} (\check{x}^s_0) = \frac{\partial u}{\partial n} (\check{x}^s_0)$. Let us assume that the first possibility (i) occurs. Then for $\delta$ small we define a function $u_1^{\delta} = u_1 -\delta$. If the second possibility (ii) occurs then we define $u_2^{\delta} = u_2 +\delta$. By the same reason as above, $u_1^{\delta}$ and $u_2^{\delta}$ are lower and upper solutions of (\ref{1.1}), respectively, and they are still non-ordered if $\delta$ is small enough. Moreover, for $\mathcal M_2^{\delta}$ defined by means of $u_1^{\delta}, u_2^{\delta}$, we have that $u\notin \overline {\mathcal M}^{\delta}_2$. By Lemma \ref{lem2.4} there must be $\check{u} \in \overline {\mathcal M}^{\delta}_2$ which is a solution of (\ref{1.1}) and $\check{u}\neq u$. This contradicts again the uniqueness of $u$. The proof of multiplicity result stated in Theorem \ref{th1.3} (ii) is thus proved and so the whole proof is finished.\hfill$\square$ \paragraph{Acknowledgements.} This research was partially supported by grant number 201/00/0376 from the Grant Agency of the Czech Republic and Ministry of Education of Czech Republic, number MSM 235200001. 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