\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{equation}
  \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}
  \end{equation}
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
\begin{equation}
\lambda_k:=
  \inf_{\mathcal{A}\in\mathcal{F}_k}\sup_{u\in
  \mathcal{A}}I(u).\label{eq:1_2}
\end{equation}
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{equation}
\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}
\end{equation}
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
\begin{equation}
  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}
\end{equation}
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
\begin{equation}
  \lim_{x\to \pm \infty }\frac{g(x)}{|x|^{p-1}}=0 \label{eq:1_5}
\end{equation}
and
\begin{equation}\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}\end{equation}
or
\begin{equation}\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}
\end{equation}
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
\begin{equation}
  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}
\end{equation}
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{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
which implies
\begin{equation}\|v_n\|\to \|v\| .\label{eq:3_8}\end{equation}
The uniform convexity of $W_0^{1,p}(0,\pi)$ then yields
\begin{equation}
  v_n\to v \text{ in } W_0^{1,p}(0,\pi),\quad \|v\|=1.\label{eq:3_9}
\end{equation}
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 $w\in W_0^{1,p}(0,\pi)$:
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
  \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}
\end{equation}
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
\begin{equation}
  F(t)\geq c_\varepsilon \text{ \  for any  \ }t>K,\ \ \
  F(t)\leq d_\varepsilon \text{ \ for any \ }t<-K.\label{eq:3_12}
\end{equation}
On the other hand, the continuity of $F$ on $\mathbb{R}$ implies that for
any $K>0$ there exists $c(K)>0$ such that
\begin{equation}
  |F(t)|\leq c(K) \quad \text{for any  } t\in
  \langle -K,K\rangle .\label{eq:3_13}
\end{equation}
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
\begin{equation}
  \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}
\end{equation}
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
\begin{equation}
  \lim_{t\to +\infty}J_\lambda (tu)=-\infty \quad
    \forall u\in\mathcal{A}\,.\label{eq:3_15}
\end{equation}
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
\begin{equation}
  \alpha:=\inf\{J_\lambda (u): u\in\mathcal{E}_{k+1}\}\in\mathbb{R}.\label{eq:3_16}
\end{equation}
 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 $\bar\varepsilon <c-\gamma$
and we apply Deformation Lemma. We get a deformation
$\phi$ and a corresponding
$\varepsilon >0$.
By definition of $c$ there is an $h\in\Gamma$ such that
$$
\sup_{x\in B_k} J_\lambda(h(x))<c+\varepsilon .
$$
Now when
$$ \tilde h(x):=\phi (h(x),1),$$
we obtain
$$
\tilde h\in\Gamma,\;
\forall x\in B_k: J_\lambda(\tilde h(x))=J_\lambda(\phi (h(x),1))
\leq c-\varepsilon,
$$
it is a contradiction to the definition of $c$.
\end{proof}

\begin{lemma} \label{lem4}
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
$\lambda=\lambda_{k}$.
Then there exists at least one weak solution of  \eqref{eq:1_1}.
\end{lemma}

\begin{proof}
At first we assume \eqref{eq:1_6}. Let $(\mu_n)$ be a
sequence in $(\lambda_k ,\lambda_{k+1})$ such that
$\mu_n\searrow\lambda_k $.
Now thanks to Lemma \ref{lem3}, we have:
For all $n\in\mathbb{N}$ there exists $c_n\in \mathbb{R}$ and $u_n\in W_0^{1,p}(0,\pi)$ such that
$$
J'_{\mu_n}(u_n)=0\ \text{ and }\
J_{\mu_n}(u_n)= c_n\geq \alpha_n:= \inf\{J_{\mu_n}(u):
u\in\mathcal{E}_{k+1}\}.
$$
It follows from \eqref{eq:1_5} and from the monotonousness
of $(\mu_n)$ that
for all $n\in\mathbb{N}$, all $u\in\mathcal{E}_{k+1}$ and all
$\varepsilon >0$, 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
\begin{equation}
  v\in\ker  (-\Delta _p-\lambda_k )\setminus \{ 0\}\label{eq:3_17}
\end{equation}
such that (up to subsequences)
\begin{equation}
  \frac{u_n}{\| u_n\| }\to v\quad  \text{in } W_0^{1,p}(0,\pi). \label{eq:3_18}
\end{equation}
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.

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\end{document}