\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 08, pp. 1--10.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/08\hfil Strong resonance problems] {Strong resonance problems for the one-dimensional $p$-Laplacian} \author[J. Bouchala\hfil EJDE-2005/08\hfilneg] {Ji\v{r}\'\i\ Bouchala} \address{Ji\v{r}\'\i\ Bouchala \hfill\break Department of Applied Mathematics V\v{S}B-Technical University Ostrava, Czech republic} \email{jiri.bouchala@vsb.cz} \date{} \thanks{Submitted June 21, 2004. Published January 5, 2005.} \thanks{Research supported by the Grant Agency of Czech Republic, 201/03/0671.} \subjclass[2000]{34B15, 34L30, 47J30} \keywords{$p$-Laplacian; resonance at the eigenvalues; \hfill\break\indent Landesman-Lazer type conditions} \thanks{These results were presented (without proof) on the conference Modelling 2001 (see \cite{bi:1}).} \begin{abstract} We study the existence of the weak solution of the nonlinear boundary-value problem \begin{gather*} -(|u'|^{p-2}u')'= \lambda |u|^{p-2} u +g(u)-h(x)\quad \hbox{in } (0,\pi) ,\\ u(0)=u(\pi )=0\,, \end{gather*} where $p$ and $\lambda$ are real numbers, $p>1$, $h\in L^{p'}(0,\pi )$ ($p' =\frac{p}{p-1}$) and the nonlinearity $g:\mathbb{R} \to \mathbb{R}$ is a continuous function of the Landesman-Lazer type. Our sufficiency conditions generalize the results published previously about the solvability of this problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} We consider the boundary-value problem $$\begin{gathered} -\Delta _p u= \lambda |u|^{p-2} u +g(u)-h(x)\quad \text{in } (0,\pi) ,\\ u(0)=u(\pi )=0, \end{gathered} \label{eq:1_1}$$ where $p>1$, $g: \mathbb{R} \to \mathbb{R}$ is a continuous function, $h\in L^{p'}(0,\pi )\ (p' =\frac{p}{p-1} )$, $\lambda\in\mathbb{R}$, and $-\Delta _p$ is the (one-dimensional) $p$-Laplacian, i.e. $\Delta _p u:= (|u'|^{p-2}u')'$. Problem \eqref{eq:1_1} can be thought of as a perturbation of the homogeneous eigenvalue problem \begin{gather*} -\Delta _p u= \lambda |u|^{p-2} u \quad \hbox{in } (0,\pi) ,\\ u(0)=u(\pi )=0\,. \end{gather*} We say that $\lambda\in\mathbb{R}$ is an eigenvalue of $-\Delta _p$ if there exists a function $u\in W_0^{1,p} (0,\pi )$, $u\not\equiv 0$, such that $$\int_0^\pi |u'|^{p-2}u' v' \,\text{d} x =\lambda \int_0^\pi |u|^{p-2}uv \,\text{d} x \quad \forall v\in W_0^{1,p} (0,\pi )\,.$$ The function $u$ is then called an eigenfunction of $-\Delta_p$ corresponding to the eigenvalue $\lambda$ and we write $$u\in\ker (-\Delta _p-\lambda )\setminus \{ 0\}.$$ Consider the functional $$I: W_0^{1,p}(0,\pi )\setminus\{ 0\}\to\mathbb{R} ;\quad I(u):=\frac{\int_0^\pi | u' |^p \,\text{d} x}{\int_0^\pi |u|^p\,\text{d} x}$$ and the manifold $$\mathcal{S}:= \{ u\in W_0^{1,p}(0,\pi ): {\| u\|}_{L^p (0,\pi)}=1\}.$$ For $k\in\mathbb{N}$ let $$\mathcal{F}_k:=\{ \mathcal{A}\subset \mathcal{S}: \text{there exists a continuous odd surjection } h: S^k\rightarrow\mathcal{A}\},$$ where $S^k$ represents the unit sphere in $\mathbb{R}^k$. Next define $$\lambda_k:= \inf_{\mathcal{A}\in\mathcal{F}_k}\sup_{u\in \mathcal{A}}I(u).\label{eq:1_2}$$ It is known that $\lambda_k$ is an eigenvalue of $-\Delta _p$ (see \cite{bi:3}) and that $(\lambda_k)$ represents complete set of eigenvalues \cite{bi:4} (For any $k\in\mathbb{N}$, $\lambda_k = \big(\frac{k\pi_p}{\pi}\big)^p$, where $\pi_p:= 2(p-1)^{\frac 1p}\int_0^1\frac{\,\text{d} s}{(1-s^p)^{\frac1p}}$). Moreover, for any $k\in\mathbb{N}$ we have $0<\lambda_{k}<\lambda_{k+1}$ and any corresponding eigenfunction has the strong unique continuation property'', i.e. $$\begin{gathered} \forall v\in\ker (-\Delta_p-\lambda_{k} ) \setminus \{ 0\},\quad \|v\|=1:\\ \left(\forall\delta > 0\right) \left(\exists \eta (\delta )>0\right): \mathop{\rm meas}\{ x\in (0,\pi): |v(x)|\leq\eta (\delta )\}<\delta. \end{gathered}\label{eq:1_3}$$ The symbol $\|\cdot\|$ indicates the norm in the Sobolev space $W_0^{1,p}(0,\pi)$, i.e. $$\|u\|=\Big(\int_0^\pi |u'|^p\,\text{d} x\Big)^{1/p}.$$ Our paper is motivated by the results in \cite{bi:2} and \cite{bi:3}. The following theorem generalizes those results for the one-dimensional problem \eqref{eq:1_1}. \begin{theorem} Let us define $$F(x):=\begin{cases}\frac{p}{x}\int_0^x g(s)\,\text{d} s -g(x),& x\neq 0,\\ (p-1)g(0),& x=0,\end{cases} \label{eq:1_4}$$ and set \begin{gather*} \overline{F(-\infty )}=\limsup_{x\to -\infty }F(x),\quad \underline{F(+\infty )}=\liminf_{x\to +\infty }F(x),\\ \overline{F(+\infty )}=\limsup_{x\to +\infty }F(x),\quad \underline{F(-\infty )}=\liminf_{x\to -\infty }F(x). \end{gather*} We suppose $$\lim_{x\to \pm \infty }\frac{g(x)}{|x|^{p-1}}=0 \label{eq:1_5}$$ and $$\begin{gathered} \forall v\in\ker (-\Delta _p-\lambda )\setminus \{ 0\}:\\ (p-1)\int_0^\pi h(x)v (x)\,\text{d} x < \underline{F(+\infty )}\int_0^\pi v^+(x) \,\text{d} x+ \overline{F(-\infty )}\int_0^\pi v^-(x)\,\text{d} x, \end{gathered}\label{eq:1_6}$$ or $$\begin{gathered} \gathered \forall v\in\ker (-\Delta _p-\lambda )\setminus \{ 0\}:\\ (p-1)\int_0^\pi h(x)v (x)\,\text{d} x > \overline{F(+\infty )}\int_0^\pi v^+(x) \,\text{d} x+ \underline{F(-\infty )}\int_0^\pi v^-(x)\,\text{d} x, \endgathered \end{gathered}\label{eq:1_7}$$ where $$v^+:=\max\{0,v\},\quad v^-:=\min\{0,v\}.$$ Then there exists at least one weak solution of the boundary-value problem \eqref{eq:1_1}; i.e. there exists $u\in W_0^{1,p}(0,\pi)$ such that for all $v\in W_0^{1,p}(0,\pi)$, $$\int_0^\pi |u'|^{p-2} u' v' \,\text{d} x =\lambda \int_0^\pi |u|^{p-2}uv \,\text{d} x +\int_0^\pi g(u)v \,\text{d} x -\int_0^\pi hv \,\text{d} x.$$ \end{theorem} Note that if $\lambda$ is not an eigenvalue of $-\Delta _p$ then the conditions \eqref{eq:1_6} and \eqref{eq:1_7} are vacuously true. \section{Preliminaries} Let $$J_{\lambda}(u):= \frac{1}{p} \int_0^\pi | u'|^p \,\text{d} x-\frac{\lambda }{p} \int_0^\pi |u|^p \,\text{d} x- \int_0^\pi G(u) \,\text{d} x+ \int_0^\pi hu \,\text{d} x,\label{eq:2_1}$$ where $$G(t):=\int_0^t g(s)\,\text{d} s.$$ It is well known that $J_\lambda\in C^1(W_0^{1,p}(0,\pi),\mathbb{R})$, and that for all $v\in W_0^{1,p}(0,\pi)$, $$\langle J_\lambda'(u),v \rangle = \int_0^\pi | u'|^{p-2} u' v' \,\text{d} x -\lambda \int_0^\pi |u|^{p-2}uv \,\text{d} x -\int_0^\pi g(u)v \,\text{d} x +\int_0^\pi hv \,\text{d} x.$$ It follows that weak solutions of \eqref{eq:1_1} correspond to critical points of $J_\lambda$. The next theorem plays a fundamental role in proving that $J_\lambda$ has critical points of saddle point type (see~\cite{bi:3,bi:5}). \begin{lemma}[Deformation Lemma] Suppose that $J_\lambda$ satisfies the Palais-Smale condition, i.e. if $(u_n)$ is a sequence of functions in $W_0^{1,p}(0,\pi)$ such that $(J_\lambda (u_n))$ is bounded in $\mathbb{R}$ and $J_\lambda '(u_n)\to 0$ in ${(W_0^{1,p}(0,\pi) )}^\ast$, then $(u_n)$ has a subsequence that is strongly convergent in $W_0^{1,p}(0,\pi)$. Let $c\in\mathbb{R}$ be a regular value of $J_\lambda$ and let $\bar\varepsilon >0$. Then there exists $\varepsilon\in (0,\bar\varepsilon )$ and a continuous one-parameter family of homeomorphisms, $\phi: W_0^{1,p}(0,\pi)\times\langle 0,1\rangle\to W_0^{1,p}(0,\pi)$, with the properties: \begin{itemize} \item[(i)] if $t=0$ or $|J_\lambda(u)-c|\geq\bar\varepsilon$, then $\phi (u,t)=u$, \item[(ii)] if $J_\lambda (u)\leq c +\varepsilon$, then $J_\lambda(\phi(u,1))\leq c -\varepsilon$. \end{itemize} \end{lemma} \section{Proof of main Theorem} The proof is divided into four lemmas. First we prove that functional $J_\lambda$ satisfies the Palais-Smale condition, and in the next steps we prove our theorem separately for situations: $\lambda<\lambda_1$, $\lambda_k<\lambda<\lambda_{k+1}$ and $\lambda=\lambda_k$. \begin{lemma} \label{lem1} Let us assume \eqref{eq:1_5} and (\eqref{eq:1_6} or \eqref{eq:1_7}). Then the functional $J_\lambda$ satisfies the Palais-Smale condition. \end{lemma} \begin{proof} We will start with the proof that any Palais-Smale sequence is bounded in $W_0^{1,p}(0,\pi)$. Suppose, by contradiction, that $(u_n)$ is a sequence of functions in $W_0^{1,p}(0,\pi)$ such that \begin{gather} (J_\lambda (u_n)) \text{ is bounded in }\mathbb{R},\label{eq:3_1} \\ J_\lambda '(u_n)\to 0 \text{ in }{(W_0^{1,p}(0,\pi) )}^\ast,\label{eq:3_2}\\ \| u_n\| \to +\infty .\label{eq:3_3} \end{gather} Due to the reflexivity of $W_0^{1,p}(0,\pi)$ and the compact embeding $$W_0^{1,p}(0,\pi) \hookrightarrow\hookrightarrow C^0(< 0,\pi >),$$ there exists $v\in W_0^{1,p}(0,\pi)$ such that (up to subsequences) \begin{gather} v_n:= \frac{u_n}{\| u_n\| } \rightharpoonup v \quad \text{(i.e., weakly) in } W_0^{1,p}(0,\pi), \label{eq:3_4}\\ v_n\to v \quad \text{ (i.e., strongly) in } C^0\big(\langle 0,\pi \rangle\big) . \label{eq:3_5} \end{gather} From \eqref{eq:3_2}, \eqref{eq:3_3} and \eqref{eq:3_4}, we have \begin{aligned} &\frac{\langle J_\lambda '(u_n),v_n-v\rangle}{\| u_n\| ^{p-1}}\\ &=\int_0^\pi | v_n' |^{p-2} v_n '(v_n-v)'\,\text{d} x- \lambda \int_0^\pi |v_n |^{p-2}v_n (v_n-v)\,\text{d} x\\ &\quad -\int_0^\pi \frac{g(u_n)}{\| u_n\|^{p-1}} (v_n-v)\,\text{d} x+ \int_0^\pi \frac{h}{\| u_n\|^{p-1}} (v_n-v)\,\text{d} x\to 0, \end{aligned} \label{eq:3_6} and since the last three terms approach 0 (here we need the assumption \eqref{eq:1_5}), we have $$\int_0^\pi | v_n' |^{p-2} v_n' (v_n-v)'\,\text{d} x\to 0.$$ It follows from here, \eqref{eq:3_4} and from the H\"older inequality that \begin{aligned} 0&\leftarrow \int_0^\pi | v_n' |^{p-2} v_n' (v_n-v)'\,\text{d} x- \int_0^\pi | v' |^{p-2} v' (v_n-v)'\,\text{d} x\\ &=\int_0^\pi |v_n' |^{p}\,\text{d} x-\int_0^\pi | v_n '|^{p-2} v_n' v'\,\text{d} x-\int_0^\pi | v' |^{p-2} v' v_n'\,\text{d} x+ \int_0^\pi | v' |^{p}\,\text{d} x \\ &\geq \|v_n\|^{p}- \|v_n\|^{p-1}\|v\|- \|v\|^{p-1}\|v_n\|+\|v\|^{p}\\ &=(\|v_n\|^{p-1}-\|v\|^{p-1})(\|v_n\|-\|v\| )\geq 0 \end{aligned}\label{eq:3_7} which implies $$\|v_n\|\to \|v\| .\label{eq:3_8}$$ The uniform convexity of $W_0^{1,p}(0,\pi)$ then yields $$v_n\to v \text{ in } W_0^{1,p}(0,\pi),\quad \|v\|=1.\label{eq:3_9}$$ It follows from \eqref{eq:3_2} and \eqref{eq:3_3} that, for any $w\in W_0^{1,p}(0,\pi)$, \begin{align*} \frac{\langle J_\lambda '(u_n),w\rangle}{\| u_n\| ^{p-1}} &= \int_0^\pi | v_n' |^{p-2} v_n' w'\,\text{d} x- \lambda \int_0^\pi |v_n |^{p-2}v_n w\,\text{d} x\\ &\quad -\int_0^\pi \frac{g(u_n)}{\| u_n\|^{p-1}} w\,\text{d} x+ \int_0^\pi \frac{h}{\| u_n\|^{p-1}} w\,\text{d} x\to 0. \end{align*} Now the last two terms approach zero. Hence for all $w\in W_0^{1,p}(0,\pi)$: $$\int_0^\pi | v_n' |^{p-2} v_n' w'\,\text{d} x- \lambda \int_0^\pi |v_n |^{p-2}v_n w\,\text{d} x\to 0.\label{eq:3_10}$$ It is known \cite{bi:3} that the maps $A,\ B: W_0^{1,p}(0,\pi)\to \big( W_0^{1,p}(0,\pi)\big)^\ast$; $$\langle Au,w\rangle:= \int_0^\pi | u' |^{p-2} u' w'\,\text{d} x,\quad \langle Bu,w\rangle:= \int_0^\pi |u |^{p-2}u w\,\text{d} x$$ are continuous, and therefore from \eqref{eq:3_9} and \eqref{eq:3_10} we have $$\int_0^\pi |v' |^{p-2} v' w'\,\text{d} x= \lambda \int_0^\pi |v |^{p-2}v w\,\text{d} x, \quad \forall w\in W_0^{1,p}(0,\pi )$$ and $$v\in\text{ker} (-\Delta _p-\lambda )\setminus \{ 0\},\quad \| v\|=1.$$ The boundedness of $(J_\lambda (u_n))$, $J_\lambda '(u_n)\to 0$, and $\| u_n\|\to\infty$ imply \begin{align*} \frac{\langle J_\lambda '(u_n),u_n\rangle - pJ_\lambda (u_n)}{\| u_n\|} &= \int_0^\pi\frac{pG(u_n)-g(u_n)u_n}{\| u_n\|}\,\text{d} x- (p-1)\int_0^\pi h\frac{u_n}{\| u_n\|}\,\text{d} x\\ &=\int_0^\pi F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x - (p-1)\int_0^\pi h\frac{u_n}{\| u_n\|}\,\text{d} x\to 0 . \end{align*} Hence $$\lim\int_0^\pi F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x = (p-1)\int_0^\pi hv\,\text{d} x. \label{eq:3_11}$$ Now we assume \eqref{eq:1_6} (the other case \eqref{eq:1_7} is treated similarly). It follows $$\underline{F(+\infty )}>-\infty \quad \text{and}\quad \overline{F(-\infty )}<+\infty .$$ For arbitrary $\varepsilon >0$ set \begin{gather*} c_\varepsilon:=\begin{cases} \underline{F(+\infty )}-\varepsilon \ \text{ if }\ \underline{F(+\infty )}\in\mathbb{R} ,\\ \frac{1}{\varepsilon}\text{ if }\underline{F(+\infty )}=+\infty ; \end{cases} \\[2pt] d_\varepsilon:=\begin{cases} \overline{F(-\infty )}+\varepsilon \ \text{ if }\ \overline{F(-\infty )}\in\mathbb{R} ,\\ -\frac{1}{\varepsilon}\ \text{ if }\ \overline{F(-\infty )}=-\infty . \end{cases} \end{gather*} Then for any $\varepsilon>0$ there exists $K>0$ such that $$F(t)\geq c_\varepsilon \text{ \ for any \ }t>K,\ \ \ F(t)\leq d_\varepsilon \text{ \ for any \ }t<-K.\label{eq:3_12}$$ On the other hand, the continuity of $F$ on $\mathbb{R}$ implies that for any $K>0$ there exists $c(K)>0$ such that $$|F(t)|\leq c(K) \quad \text{for any } t\in \langle -K,K\rangle .\label{eq:3_13}$$ Let us choose $\varepsilon>0$ and consider the corresponding $K>0$ and $c(K)>0$ given by \eqref{eq:3_12} and \eqref{eq:3_13}, respectively. Set $$\int_0^\pi F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x= A_{K,n}+B_{K,n}+C_{K,n}+D_{K,n}+E_{K,n},\label{eq:3_14}$$ where \begin{gather*} A_{K,n}=\int_{\{\scriptstyle x\in (0,\pi ) :\atop |u_n(x)|\leq K\}} F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x,\quad B_{K,n}=\int_{\{\scriptstyle x\in (0,\pi ) :\atop {u_n(x)> K,\atop v(x)>0\}}} F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x, \\ C_{K,n}=\int_{\{\scriptstyle x\in (0,\pi ) :\atop { u_n(x)> K,\atop v(x)\leq 0\}}} F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x,\quad D_{K,n}=\int_{\{\scriptstyle x\in (0,\pi ) :\atop {u_n(x)< -K,\atop v(x)< 0\}}} F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x,\\ E_{K,n}=\int_{\{\scriptstyle x\in (0,\pi ) :\atop { u_n(x)< -K,\atop v(x)\geq 0\}}} F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x\,. \end{gather*} Before estimating these integrals we claim that for any $K>0$ the following assertions are true: \begin{gather*} \lim_{n\to\infty}\mathop{\rm meas}\{ x\in (0,\pi ): \ u_n(x)> K \text{ and } v(x)\leq 0\} =0 ,\\ \lim_{n\to\infty}\ \mathop{\rm meas}\{ x\in (0,\pi ): \ u_n(x)< - K \text{ and } v(x)\geq 0\}=0 ,\\ \lim_{n\to\infty}\mathop{\rm meas}\{ x\in (0,\pi ): \ u_n(x)\leq K \text{ and } v(x)> 0\} =0 ,\\ \lim_{n\to\infty}\mathop{\rm meas}\{ x\in (0,\pi ): u_n(x)\geq - K \text{ and } v(x)< 0\} =0 \end{gather*} cf. \eqref{eq:1_3} and \eqref{eq:3_5}. We are now ready to estimate the integrals from \eqref{eq:3_14}. \begin{align*} |A_{K,n}|& \leq \frac{c(K) K\pi }{\|u_n\|}\to 0,\\ B_{K,n} & \geq c_\varepsilon ( \smallint_{\{\scriptstyle x\in (0,\pi ) :\atop v(x)>0\}} v_n\,\text{d} x - \smallint_{\{\scriptstyle x\in (0,\pi ) :\atop {u_n(x)\leq K,\atop v(x)>0\}}} v_n\,\text{d} x )\to c_\varepsilon \smallint_{\{\scriptstyle x\in (0,\pi ) :\atop v(x)>0\}} v(x)\,\text{d} x ,\\ C_{K,n} & \geq c_\varepsilon \smallint_{\{\scriptstyle x\in (0,\pi ) :\atop { u_n(x)> K,\atop v(x)\leq 0\}}} v_n\,\text{d} x\to 0,\\ D_{K,n} & \geq d_\varepsilon (\smallint_{\{\scriptstyle x\in (0,\pi ) :\atop v(x)<0\}} v_n\,\text{d} x - \smallint_{\{\scriptstyle x\in (0,\pi ) :\atop {u_n(x)\geq -K,\atop v(x)<0\}}} v_n\,\text{d} x )\to d_\varepsilon \smallint_{\{\scriptstyle x\in (0,\pi ) :\atop v(x)<0\}} v(x)\,\text{d} x ,\\ E_{K,n} & \geq d_\varepsilon \smallint_{\{\scriptstyle x\in (0,\pi) :\atop{ u_n(x)<- K,\atop v(x)\geq 0\}}} v_n\,\text{d} x\to 0. \end{align*} Hence (see \eqref{eq:3_14}), for any $\varepsilon >0$, \begin{align*} \liminf\int_0^\pi F(u_n)\frac{u_n}{\| u_n\|}\,\text{d} x &= \liminf\ (A_{K,n}+B_{K,n}+C_{K,n}+D_{K,n}+E_{K,n})\\ &\geq c_\varepsilon \int_{\{\scriptstyle x\in (0,\pi ) :\atop v(x)>0\}} v(x)\,\text{d} x + d_\varepsilon \int_{\{\scriptstyle x\in (0,\pi ) :\atop v(x)<0\}} v(x)\,\text{d} x, \end{align*} which together with \eqref{eq:3_11} implies $$(p-1)\int_0^\pi h(x)v (x)\,\text{d} x \geq \underline{F(+\infty )}\int_0^\pi v^+(x) \,\text{d} x+ \overline{F(-\infty )}\int_0^\pi v^-(x)\,\text{d} x,$$ contradicting \eqref{eq:1_6}. This proves that $(u_n)$ is bounded. The rest of the proof is very easy. If the sequence $(u_n)$, which is bounded in $W_0^{1,p}(0,\pi)$, satisfies conditions \eqref{eq:3_1} and \eqref{eq:3_2}, then there exists $u\in W_0^{1,p}(0,\pi)$ such that (passing to subsequences) $$u_n\rightharpoonup u \text{ in }W_0^{1,p}(0,\pi),\quad u_n\to u \text{ in }C^0(\langle 0,\pi \rangle).$$ It follows from here, \eqref{eq:3_2} and \eqref{eq:1_5} that \begin{align*} \lim\langle J_\lambda '(u_n),u_n-u\rangle &=\lim \int_0^\pi | u_n' |^{p-2} u_n '(u_n-u)'\,\text{d} x -\lambda \int_0^\pi |u_n |^{p-2}u_n (u_n-u)\,\text{d} x \\ &\quad -\int_0^\pi g(u_n) (u_n-u)\,\text{d} x+ \int_0^\pi h (u_n-u)\,\text{d} x\\ &=\lim\int_0^\pi | u_n' |^{p-2} u_n '(u_n-u)'\,\text{d} x=0 \end{align*} which implies $\| u_n \| \to \| u \|$ (cf. \eqref{eq:3_7}). The uniform convexity of $W_0^{1,p}(0,\pi)$ then yields $u_n\to u\text{ in }W_0^{1,p}(0,\pi )$. The proof is complete. \end{proof} \begin{lemma} \label{lem2} Let us assume \eqref{eq:1_5} and let $\lambda <\lambda_1$. Then there exists at least one weak solution of \eqref{eq:1_1}. \end{lemma} \begin{proof} Assumption \eqref{eq:1_5} and the variational characterization of $\lambda_1$ yield: For all $u\in W_0^{1,p}(0,\pi)$ and all $\varepsilon>0$ there exists $c>0$ such that \begin{align*} J_\lambda(u)&= \frac{1}{p} \int_0^\pi | u'|^p \,\text{d} x-\frac{\lambda _1}{p} \int_0^\pi |u|^p \,\text{d} x + \frac{\lambda _1-\lambda}{p} \int_0^\pi |u|^p \,\text{d} x\\ &\quad -\int_0^\pi G(u) \,\text{d} x+ \int_0^\pi hu \,\text{d} x\\ &\geq \frac{\lambda _1-\lambda}{p} \int_0^\pi |u|^p \,\text{d} x- c\int_0^\pi |u| \,\text{d} x- \frac{\varepsilon}{p}\int_0^\pi |u|^p \,\text{d} x- \int_0^\pi |hu| \,\text{d} x\\ &\geq \frac{\lambda _1-\lambda-\varepsilon}{p} \| u\| _{L^p (0,\pi )}^p - c\| u\| _{L^1(0,\pi )}- \| h\| _{L^{p'}(0,\pi )}\| u\| _{L^p(0,\pi )}. \end{align*} Hence the functional $J_\lambda$ is bounded from bellow on $W_0^{1,p}(0,\pi)$. It follows from this and from Lemma \ref{lem1} that $J_\lambda$ attains its global minimum on $W_0^{1,p}(0,\pi)$ \cite[Corollary 2.5]{bi:6}. \end{proof} \begin{lemma} \label{lem3} Let us assume \eqref{eq:1_5} and (\eqref{eq:1_6} or \eqref{eq:1_7}). Let there exists $k\in\mathbb{N}$ such that $\lambda_k <\lambda<\lambda_{k+1}$. Then there exists at least one weak solution of \eqref{eq:1_1}. \end{lemma} \begin{proof} Let $m\in (\lambda_k ,\lambda )$ and let $\mathcal{A}\in\mathcal{F}_k$ be such that $$\sup_{u\in\mathcal{A}}I(u)\leq m$$ (see Section 1 for $\mathcal{F}_k$). Then (we again need \eqref{eq:1_5}): For all $u\in\mathcal{A}$, all $t>0$ and all $\varepsilon>0$, there exists $c>0$ such that \begin{align*} J_\lambda(tu) &=\frac{1}{p} t^p \Big(\int_0^\pi |u'|^p \,\text{d} x-\lambda \int_0^\pi |u|^p \,\text{d} x\Big) - \int_0^\pi G(tu) \,\text{d} x+t\int_0^\pi hu \,\text{d} x\\ &\leq \frac{1}{p} t^p( m-\lambda)\| u\|_{L^p(0,\pi )}^p +c t\|u\|_{L^1(0,\pi )}\\ &\quad +\frac{\varepsilon}{p}t^p\|u\|_{L^p(0,\pi )}^p +t\|h\|_{L^{p'}(0,\pi )}\|u\|_{L^p(0,\pi )}\\ &= \frac{1}{p} t^p ( m-\lambda +\varepsilon )\|u\|_{L^p(0,\pi )}^p +t \big( c\|u\|_{L^1(0,\pi )}+\|h\|_{L^{p'}(0,\pi )}\|u\|_{L^p(0,\pi )}\big), \end{align*} and $$\lim_{t\to +\infty}J_\lambda (tu)=-\infty \quad \forall u\in\mathcal{A}\,.\label{eq:3_15}$$ Now we continue similarly as in~\cite{bi:3}. Let $$\mathcal E_{k+1}:= \{ u\in W_0^{1,p}(0,\pi): \int_0^\pi | u' |^{p}\,\text{d} x\geq\lambda _{k+1} \int_0^\pi | u |^{p}\,\text{d} x\},$$ and notice that for all $u\in\mathcal{E}_{k+1}$, all $\varepsilon>0$ there exists $c>0$ such that $$J_\lambda (u)\geq \frac{1}{p} \left(\lambda_{k+1}-\lambda -\varepsilon\right) \| u\|_{L^p(0,\pi )}^p -c\| u\|_{L^1(0,\pi )}-\|h\|_{L^{p'}(0,\pi )}\|u\|_{L^p(0,\pi)}.$$ Hence $$\alpha:=\inf\{J_\lambda (u): u\in\mathcal{E}_{k+1}\}\in\mathbb{R}.\label{eq:3_16}$$ From \eqref{eq:3_15} and \eqref{eq:3_16} we see that there exists $T>0$ such that $$\gamma:=\max\{J_\lambda (tu): u\in\mathcal{A}\ \text{ and }\ t\in \left\langle T,+\infty \right)\}< \alpha .$$ The rest of the proof can be copied from \cite{bi:3}. If we define \begin{gather*} T\mathcal{A}:=\{ tu\in W_0^{1,p}(0,\pi): u\in\mathcal{A}\text{ and } t\in\left< T,+\infty\right)\},\\ \Gamma:=\{ h\in C^0 ( B_k, W_0^{1,p}(0,\pi)): h |_{S^k} \text{ is an odd map into } T\mathcal{A}\},\\ \end{gather*} where $$B_k:=\{ x=(x_1,\dots ,x_k)\in\mathbb{R}^k: \|x\|_{\mathbb{R}^k}= \sqrt{x_1^2+\dots +x_k^2}\leq 1\} ,$$ then we can prove that $\Gamma$ is nonempty and if $h\in\Gamma$ then $h(B_k)\cap \mathcal{E}_{k+1}\neq\emptyset$. Moreover, from Deformation Lemma then follows that $$c:=\inf_{h\in\Gamma}\sup_{x\in B_k} J_\lambda(h(x))$$ is a critical value of $J_\lambda$. Indeed, assume by contradiction, that $c$ is a regular value of $J_\lambda$. It is clear that $c\geq \alpha$. Now we consider arbitrary $\bar\varepsilon >0$ such that $\bar\varepsilon 0$. By definition of $c$ there is an $h\in\Gamma$ such that \sup_{x\in B_k} J_\lambda(h(x))0, there exists c>0 such that \begin{align*} &J_{\mu_n}(u)\\ &\geq\frac{1}{p} (\lambda_{k+1}-\mu_n)\|u\|_{L^p(0,\pi)}^p -c\|u\|_{L^1(0,\pi)} -\frac{\varepsilon}{p}\|u\|_{L^p(0,\pi)}^p- \|h\|_{L^{p'}(0,\pi)}\|u\|_{L^p(0,\pi)}\\ &\geq \frac{1}{p} (\lambda_{k+1}-\mu_1-\varepsilon )\|u\|_{L^p(0,\pi)}^p -c\|u\|_{L^1(0,\pi)}- \|h\|_{L^{p'}(0,\pi)}\|u\|_{L^p(0,\pi)}, \end{align*} and so the sequence (c_n) is bounded below. Now we prove that the corresponding sequence of critical points, (u_n), is bounded. Suppose, by contradiction, that \|u_n\|\to +\infty. Then we can assume that there exists $$v\in\ker (-\Delta _p-\lambda_k )\setminus \{ 0\}\label{eq:3_17}$$ such that (up to subsequences) $$\frac{u_n}{\| u_n\| }\to v\quad \text{in } W_0^{1,p}(0,\pi). \label{eq:3_18}$$ Because (c_n) is bounded from below, it follows from \eqref{eq:1_6}, \eqref{eq:3_17} and \eqref{eq:3_18} that \begin{align*} 0&\leq \liminf \frac{p c_n}{\|u_n\|}\\ &\leq \limsup \frac{p c_n}{\|u_n\|}\\ &=\limsup\frac{ pJ_{\mu_n}(u_n)-\langle J'_{\mu_n}(u_n),u_n\rangle }{\|u_n\|}\\ &=\limsup \Big( -\frac{p\int_0^\pi G(u_n) \,\text{d} x -\int_0^\pi g(u_n)u_n \,\text{d} x}{\| u_n\|}+ (p-1)\int_0^\pi h\frac{u_n}{\| u_n\|}\,\text{d} x\Big)\\ &=-\liminf \Big( \frac{p\int_0^\pi G(u_n) \,\text{d} x -\int_0^\pi g(u_n)u_n \,\text{d} x}{\| u_n\|}\Big)+ (p-1)\int_0^\pi hv\,\text{d} x\\ &=-\liminf \Big( \int_0^\pi F(u_n) \frac{u_n}{\| u_n\|} \,\text{d} x\Big)+ (p-1)\int_0^\pi hv\,\text{d} x <0. \end{align*} This is a contradiction, therefore (u_n) is bounded. Thus there will be a subsequence of critical points that converges to the desired solution. Now we assume \eqref{eq:1_7}. Because for \lambda=\lambda_1 our assertion was proved in \cite{bi:2}, we focus on k>1. Let (\mu_n) be a sequence in (\lambda_{k-1} ,\lambda_{k}) such that \mu_n\nearrow\lambda_k . We can find (similarly as in \cite{bi:3}) a sequence (u_n) of critical points associated with the functionals J_{\mu_n} such that the sequence c_n:=J_{\mu_n}(u_n) is decreasing, i.e. J'_{\mu_n}(u_n)=0,\quad J_{\mu_n}(u_n)= c_n\geq c_{n+1}.  Now we are going to prove that $(u_n)$ is bounded. Suppose, by contradiction, $\|u_n\|\to\infty$. Then there exists $v\in\ker (-\Delta _p-\lambda_k )\setminus \{ 0\}$ such that (up to subsequence) $\frac{u_n}{\|u_n\| }\to v$ and \begin{align*} 0&\geq \limsup \frac{p c_n}{\|u_n\|}\geq \liminf \frac{p c_n}{\|u_n\|}\\ &=\liminf\frac{ pJ_{\mu_n}(u_n)-\langle J'_{\mu_n}(u_n),u_n\rangle }{\|u_n\|}\\ &=\liminf \Big( -\frac{p\int_0^\pi G(u_n) \,\text{d} x -\int_0^\pi g(u_n)u_n \,\text{d} x}{\| u_n\|}+ (p-1)\int_0^\pi h\frac{u_n}{\| u_n\|}\,\text{d} x\Big) \\ &=-\limsup \Big( \frac{p\int_0^\pi G(u_n) \,\text{d} x -\int_0^\pi g(u_n)u_n \,\text{d} x}{\| u_n\|}\Big)+(p-1)\int_0^\pi hv\,\text{d} x \\ &=-\limsup \Big( \int_0^\pi F(u_n) \frac{u_n}{\| u_n\|} \,\text{d} x\Big)+ (p-1)\int_0^\pi hv\,\text{d} x >0, \end{align*} which is a contradiction. Now it is a simple matter to show that, by passing to a subsequence, we obtain a critical point of $J_{\lambda_k}$ in the limit. \end{proof} \subsection*{Acknowledgements} The author is indebted to Professor Pavel Dr\'abek for his valuable comments. \begin{thebibliography}{0} \bibitem{bi:1} J. Bouchala: \emph{Resonance problems for $p$-Laplacian}, Mathematics and Computers in Simulation 61 (2003), 599--604. \bibitem{bi:2} J. Bouchala and P. Dr\'{a}bek: \emph{Strong resonance for some quasilinear elliptic equations}, Journal of Mathematical Analysis and Applications 245 (2000), 7--19. \bibitem{bi:3} P. Dr\'{a}bek and S. B. Robinson: \emph{Resonance Problems for the $p$-Laplacian}, Journal of Functional Analysis 169 (1999), 189--200. \bibitem{bi:4} M. Cuesta: \emph{On the Fu\v{c}\'{\i}k Spectrum of the Laplacian and the $p$-Laplacian}, Proceedings of Seminar in Differential Equations, University of West Bohemia in Pilsen (2000), 67--96. \bibitem{bi:5} M. 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