\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 33, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/33\hfil Multiple solutions]
{Multiple solutions for semilinear elliptic equations
with nonlinear boundary conditions}

\author[J. Harada, M. \^Otani\hfil EJDE-2012/33\hfilneg]
{Junichi Harada, Mitsuharu \^Otani}  % in alphabetical order

\dedicatory{Dedicated to the memory of Professor Riichi Iino}

\address{Junichi Harada \newline
 Department of Applied Physics,
 School of Science  and Engineering,
 Waseda University, 3-4-1 Okubo,
 Shinjuku-ku Tokyo, 169-8555, Japan}
\email{harada-j@aoni.waseda.jp}

\address{Mitsuharu \^Otani \newline
Department of Applied Physics,
 School of Science  and Engineering,
 Waseda University, 3-4-1 Okubo,
Shinjuku-ku Tokyo, 169-8555, Japan}
\email{otani@waseda.jp}

\thanks{Submitted November 17, 2011. Published February 23, 2012.}
\thanks{M. \^Otani was supported by grant 21340032
from the Ministry of Education, Culture, \hfill\break\indent 
Sports, Science and Technology, Japan}

\subjclass[2000]{35J20}
\keywords{Nonlinear boundary conditions}

\begin{abstract}
 We consider the elliptic problem with nonlinear boundary conditions:
 \begin{gather*}
 -\Delta u +bu=f(x,u)\quad\text{in }\Omega,\\
 -\partial_{\nu}u=|u|^{q-1}u-g(u)\quad\text{on }\partial\Omega,
 \end{gather*}
 where  $\Omega$ is a bounded domain in $\mathbb{R}^n$.
 Proving the existence of solutions of this problem relies
 essentially on a variational argument.
 However, since $L^{q+1}(\partial\Omega)\subset H^1(\Omega)$
 does not hold for large $q$, the standard variational method can not be
 applied directly. To overcome this difficulty, we use approximation methods
 and uniform a priori estimates for solutions of approximate equations.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

We consider the heat equations with nonlinear boundary conditions of the form:
\begin{equation}
\begin{gathered}
 u_t = \Delta u +bu, \quad (x,t)\in\Omega\times(0,T),\\
 -\partial_{\nu}u = \beta(u), \quad  (x,t)\in\partial\Omega\times(0,T),\\
 u(x,0) = u_0(x), \quad  x\in\Omega,
\end{gathered} \label{eP}
\end{equation}
where $\partial_{\nu}$ denotes the outward normal derivative on the boundary,
which appears in  models describing diffusion systems governed
by some radiation law on the boundary.
The standard boundary conditions for heat equations are usually assumed to
be Dirichlet-type, Neumann-type or mixed-type boundary conditions.
 This convention could be meaningful when the total system on the boundary is
 controlled so as to keep the prescribed boundary conditions.
However when the whole system is very large, it would be no more possible to control
the flux of heat through the boundary.
For such a case, the boundary condition is expected to be posed by considering
the heat radiation law on the boundary.
The typical example of this kind of radiation law on the boundary is derived
from the so-called Stefan-Boltzmann's radiation law,
which says that the heat energy radiation  from the surface of the body $J$ is given by
$J=\sigma(T^4-T_s^4)$,
where $\sigma>0$ is a physical constant,
$T$ is the surface temperature and $T_s$ is the outside temperature.
Thus Stefan-Boltzmann's law gives an example
where $\beta(u)$ is a monotone increasing function.
For this case,
the unique solvability for parabolic equations\eqref{eP} is completely covered by
the abstract (subdifferential operator)
theory by Br\'ezis \cite{Brezis1}.

However, Stefan-Boltzmann's radiation law is valid only for an
idealized situation, in other words, the radiation law rulling real
situations might be perturbed from Stefan-Boltzmann's law.
In particular,
if we consider the case where the heat flux radiated from the surface is
reflected by its surrounding materials,
then we must consider also the absorption effect.
For such a case,
$\beta(u)$ could not be a monotone increasing function anymore,
but monotone increasing with small perturbation; i.e.,
the boundary condition should be altered by
\[
 -\partial_{\nu}u=\beta(u)-g(u) \quad x\in\partial\Omega,
\]
where $\beta(u)$ is a monotone increasing function and $g(u)$ is its perturbation.
In fact,
such a kind of non-monotone radiation-absorption models are already
proposed from the view point of engineering (see e.g. \cite{Davies}).

In this article,
we are concerned with such non-monotone radiation-absorption models
and study the stationary problem of a general form:
\begin{equation}\label{non-bon}
\begin{gathered}
 -\Delta u + bu = f(x,u) \quad \text{in } \Omega, \\
 -\partial_\nu u = \beta(u) - g(u) \quad  \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $b>0$ and $\Omega\subset {\mathbb{R}}^N$
is a bounded domain with smooth boundary $\partial\Omega$.
In \cite{Harada-H-O},
the existence and the $H^2$-regularity of solutions of \eqref{non-bon}
is studied for the special case $f(x,u)=f(x)$
under the following conditions on $\beta(u)$ and $g(u)$.

\begin{itemize}
\item[(A1)] $\beta(u)$ is a continuous and monotone increasing function,

\item[(A2)]  $\lim_{|u|\to\infty}\beta(u)/u=\infty$,

\item[(A3)]  $g(u)$ is a locally Lipschitz continuous function
and there exist $\theta\in (0,1)$ and $c_1>0$ such that 
  $|  g'(u)   |\leq \theta   \beta'(u)+c_1
\quad  \forall u \in \mathbb{R}^1$,

\item[(A4)] there exists $c_2 >0$ such that
       $ |   u   \beta(u)  |\leq c_2   (   j(u) + u^2 + 1   )$
 for all $u \in \mathbb{R}^1$,
where $j(u)=  \int_0^u  \beta(s)   ds$.
\end{itemize}
The following results were presented in \cite{Harada-H-O}.

\begin{theorem}\label{linear}
Let {\rm (A1)--(A3)} be satisfied and let $f(x,u) = f(x) \in L^2(\Omega)$.
Then there exists a solution $u\in H^2(\Omega)$ of \eqref{non-bon}.
Moreover there exists $c>0$ such that
every solution $u$ of \eqref{non-bon} belonging to $ H^2(\Omega)$ satisfies
\begin{equation}\label{H2ju}
 \|   u    \|_{H^2(\Omega)}
\leq  c   \bigl(    1+ \|f\|_{L^2(\Omega)}   \bigr).
\end{equation}
\end{theorem}

Furthermore the elliptic estimates for weak solutions of \eqref{non-bon} is
also shown in \cite{Harada-H-O}.

A function $ u \in \{   u \in H^1(\Omega) ;  \beta(u), \ g(u) \in L^1(\partial\Omega)   \}$
is said to be a weak solution of \eqref{non-bon}
if $u$ satisfies
\begin{equation}\label{def:weak}
 \int_{\Omega}
 \bigl(    \nabla u \cdot \nabla \phi + b   u   \phi   \bigr)  \, dx
+ \int_{\partial\Omega} \bigl(   \beta(u)-g(u)   \bigr)   \phi \,  d\sigma
= \int_{\Omega}  f   \phi \,  dx
\end{equation}
for any $\phi\in H^1(\Omega)\cap L^{\infty}(\Omega)$.
We set $j(u) = \int_0^u \beta(s) \,  ds$ and
\[
 D(j) = \{   u \in H^1(\Omega)   ;   j(u) \in L^1(\partial\Omega)   \}.
\]

\begin{theorem}\label{th:es}
Let {\rm (A1)--(A4)} be satisfied and let $f(x,u) = f(x) \in L^2(\Omega)$.
Then every weak solution $u$ of  \eqref{non-bon} with $u \in D(j)$ satisfies
\eqref{H2ju}.
\end{theorem}

In this article, we consider the case where $f(x,u)$ satisfies the following conditions.

\begin{itemize}
\item[(B1)]   $f(x,t) \in C(\bar{\Omega}\times \mathbb{R}^1; \mathbb{R}^1)$,

\item[(B2)]  thee exist $p \in (1, 2^*)$ and  $c>0$ such that
     $ |   f(x,u)   | \leq c   (   1+ |u|^{p-1}  )$,
 where 
$$ 
2^* =\begin{cases}  \infty &\text{if $n=1$ or $2$}, \\
   \frac{2 n}{n-2} &\text{if  $n\geq 3$}.
\end{cases}
$$
\item[(B3)]  $\lim_{u\to0}\frac{f(x,u)}{u} = 0$ uniformly on $ x \in \Omega$,

\item[(B4)] there exist $\mu > 2$ and $r>0$ such that
        $ 0 < \mu  F(x,u) \leq u   f(x,u)$ for $|u|\geq r$,
where $F(x,u)=\int_0^u  f(x,s)ds$.
\end{itemize}
A typical example of a function satisfying(B1)--(B4) is
 $f(x,u)= a(x)   |u|^{p-2}u$ with $a(\cdot) \in L^\infty(\Omega)$ and
 $1<p< 2^*$.

The function $\beta(u)$ is assumed to have the power nonlinearity
$\beta(u)=|u|^{q-2}u$; i.e., we are concerned with
the equation
\begin{equation}\label{Mono5-eq}
\begin{gathered}
 -\Delta u + b   u = f(x,u) \quad \text{in } \Omega, \\
 -\partial_{\nu}u = |u|^{q-2}u - g(u) \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
We further impose the following conditions on $g(u)$.
\begin{itemize}
\item[(A5)]    $\lim_{u \to 0} \frac{g(u)}{u} = 0$.

\item[(A6)] $g(u)$ is a continuous function and
 for any $\varepsilon>0$ there exists
 $c_{\varepsilon}>0$ such that
\[
|   g(u)   | \leq \varepsilon    |u|^{q-1} + c_{\varepsilon}
   \quad  \forall u \in \mathbb{R}^1.
\]
\end{itemize}
Then our existence results are stated a follows.

\begin{theorem}\label{exist-thm}
Let {\rm(B1)--(B4)}, {\rm(A5)} and {\rm(A6)} be satisfied and let $2<q<\mu $.
Then there exists a nontrivial weak solution $u$ of \eqref{Mono5-eq}
belonging to $H^1(\Omega)\cap L^{\infty}(\Omega)$.
\end{theorem}

\begin{theorem}\label{many-thm}
Let the assumptions in Theorem  \ref{exist-thm} be satisfied and
let $f(x,u)$ and $g(u)$ be odd in $u$.
Then there exist infinitely many weak solutions
$\{u_k\}_{k=1}^{\infty}$ of \eqref{Mono5-eq}
in $H^1(\Omega) \cap L^{\infty}(\Omega)$ satisfying
\[
 \lim_{k\to \infty} I(u_k)=\infty.
\]
 Here $I(u)$ is a functional
associated with \eqref{Mono5-eq} defined by
\begin{equation}
  I(u)
= \int_{\Omega}  \Big(
 \frac{1}{2} \left(   |\nabla u|^2 + b   u^2    \right)-F(x,u) \Big)\,dx
+ \int_{\partial \Omega}
 \Big( \frac{1}{q}|u|^{q}-G(u) \Big)\, d\sigma, \label{Mono6-eq}
\end{equation}
where $G(u)=\int_0^u g(s) \, ds$.
\end{theorem}

\section{Proofs of main theorems}

\begin{proof}[Proof of Theorem \ref{exist-thm}]
We rely on the variational approach (mountain pass lemma) to prove
the existence of nontrivial solutions of \eqref{Mono5-eq}.
However
the functional $I(u)$ given by \eqref{Mono6-eq} may not be well defined
on $H^1(\Omega)$ in general,
since the functions appearing in the boundary integral might not
be integrable for any $u\in H^1(\Omega)$.
To cope with this difficulty,
we first introduce the following approximation problems:
\begin{equation}\label{Mono36-eq}
\begin{gathered} 
-\Delta u + b   u = f(x,u) \quad \text{in } \Omega, \\
-\partial_{\nu}u = \beta_k(u) - g_k(u) \quad \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where the approximation functions $\beta_k$ and $g_k$ for $\beta$ and $g$
are given by
\begin{equation*}
\beta_k(u)
=\begin{cases}
 k^{q-1}   & u>k,  \\
|u|^{q-2}u  & |u|\leq k,   \\
-k^{q-1}    & u<-k,
\end{cases}
\quad
g_k(u)
=\begin{cases}
g(k)  &u>k,   \\
g(u)  &|u|\leq k,   \\
g(-k) & u<-k.
\end{cases}
\end{equation*}
Then the functional $I_k$ associated with \eqref{Mono36-eq} is 
\[
 I_k(u) =
 \int_{\Omega}\frac{1}{2}\left(   |\nabla u|^2 + b  u^2   \right)\, dx
+ \int_{\partial\Omega}\left(   j_k(u) - G_k(u)   \right) \,d\sigma
- \int_{\Omega}F(x,u) \, dx,
\]
where $j_k(u)=\int_0^u \beta_k(s)  \, ds$, 
$G_k(u)=\int_0^u g_k(s) \, ds$.
Since $\beta_k, \ g_k\in L^{\infty}(\mathbb{R})$ and $p \in (1, 2^*)$,
it is clear that $I_k$ is well defined on $H^1(\Omega)$.
From (A6), there exists $r_0>0$ independent of $k\in\mathbb{N}$ such that
$j_k(u)-G_k(u)>0$ for $|u|> r_0$.
Hence by (A5) and the trace theorem,
for any $\eta>0$ there exists $\delta = \delta(\eta) >0$
independent of $k\in\mathbb{N}$ such that
\[
 \int_{\partial\Omega} \left(   j_k(u) - G_k(u)    \right)  d\sigma
\geq  -\eta   \|u\|_{H^1(\Omega)}^2 \quad
     \forall u \in \{   u \in H^1(\Omega)   ;
                 \|u\|_{H^1(\Omega)}<\delta   \}.
\]
Therefore, by (B3),
there exists $\mu,  \rho>0$ independent of $k \in \mathbb{N}$ such that
\begin{equation}\label{Monomu-eq}
I_k(u)\geq \mu   \|u\|_{H^1(\Omega)}^2 \quad
     \forall u \in \{   u \in H^1(\Omega)   ;
                 \|u\|_{H^1(\Omega)} = \rho   \}.
\end{equation}
Next we are going to check the (PS) condition.
We note that for $u\in H^1(\Omega)$,
\begin{align*}
 I_k(u) - \frac{(\nabla I_k(u), u)}{\mu}
&= \big(\frac{1}{2}-\frac{1}{\mu}\big)
 \int_{\Omega}  \left( |\nabla u|^2 + bu^2    \right) \,dx
-\int_\Omega\big( F(x,u)-\frac{uf(x,u)}{\mu} \big)\,dx \\
&\quad  +\int_{\partial \Omega}\big( j_k(u)-\frac{\beta_k(u)u}{\mu} \big)d\sigma
 - \int_{\partial \Omega}\big( G_k(u)-\frac{g_k(u)u}{\mu} \big)d\sigma,
\end{align*}
where $ \nabla I_k(u)$ denotes the Fr\'echet derivative of $I_k(u)$.
From (A6), for any $\eta>0$ there exists $c_\eta>0$ such that
\begin{align*}
&j_k(u)-\frac{\beta_k(u)u}{\mu}
- \big( G_k(u)-\frac{g_k(u)u}{\mu} \big)\\
&\geq
\begin{cases}
 \big(\frac{1}{q+1}-\frac{1}{\mu}\big)|u|^{q+1} -
 \left(   \eta   |u|^{q+1} + c_\eta   \right)
 & \text{if } \ |u|\leq k,\\
 \big(\frac{1}{q+1}-\frac{1}{\mu}\big)k^q|u| -
 (   \eta   k^q   |u| + c_\eta   )
& \text{if } \ |u|>k.
\end{cases}
\end{align*}
Since $1<q<\mu-1$, by choosing $\eta>0$ small enough, we deduce that
\[
 \int_{\partial \Omega}\Big( j_k(u)-\frac{\beta_k(u)   u}{\mu} \Big)d\sigma -
 \int_{\partial \Omega}\Big( G_k(u)-\frac{g_k(u)   u}{\mu} \Big)d\sigma
\geq  -c_\eta   |\partial \Omega|.
\]
Therefore by (B4), there exists a constant $C_\Omega$ depending on $ | \Omega|$ 
and $|\partial \Omega|$ such that
\begin{equation}\label{Monomi-eq}
 \big(\frac{1}{2}-\frac{1}{\mu}\big)
 \int_{\Omega}\left(   |\nabla u|^2+b   u^2   \right)dx
\leq  I_k(u)-\frac{(\nabla I_k(u), u)}{\mu}+ C_\Omega.
\end{equation}
Let $\{u_j\}_{j\in\mathbb{N}}$ be a sequence such that $I_k(u_j)\to c$ and $\nabla I_k(u_j)\to0$
in $(H^1(\Omega))^*$ as $j\to\infty$.
From \eqref{Monomi-eq},
the sequence $\{u_j\}_{j\in\mathbb{N}}$ is bounded in $H^1(\Omega)$.
Hence, there exists a subsequence of $\{u_j\}_{j\in\mathbb{N}}$ denoted again by
 $\{u_j\}_{j\in\mathbb{N}}$ which converges to $u$ weakly in $H^1(\Omega)$.

Here we note the  identity
\begin{equation}\label{est:PS}
\begin{split}
&(   \nabla I_k(u_j) - \nabla I_k(u), u_j - u   )\\
& = | \nabla ( u_j - u ) |^2_{L^2} + b   | u_j - u |^2_{L^2 }
     + \big( f(x,u_j) - f(x,u), u_j - u \big) \\
 &\quad \times \int_{\partial \Omega} ( \beta_k(u_j) - \beta_k(u), u_j - u) \,  d\sigma
     - \int_{\partial \Omega} ( g_k(u_j) - g_k(u), u_j - u)  \, d\sigma.
\end{split}
\end{equation}
Furthermore, by Rellich's compactnes theorem with (B2) and the trace theorem,
we obtain
\begin{gather*}
f(x,u_j) \to f(x,u) \quad \text{strongly in } \ L^{(p+1)/p}(\Omega),\\
\beta_k(u_j) \to \beta_k(u), \;
  g_k(u_j) \to g_k(u) \quad \text{strongly in } \ L^2(\partial \Omega).
\end{gather*}
Hence, by letting $j \to \infty$ in \eqref{est:PS}, we find that $u_j$ converges
to $u$ strongly in $H^1(\Omega)$.
Thus it is verified that $I_k$ satisfies (PS)-condition for any $k\in\mathbb{N}$.
Now we define
\begin{equation}\label{Mono38-eq}
 I_0(u) =
 \int_{\Omega}\frac{1}{2}\left( |\nabla u|^2 + b  u^2 \right)   dx
- \int_{\Omega}F(x,u)   dx.
\end{equation}
Since $I_k(u)=I_0(u)$ for $u\in C_0^\infty(\Omega)$ and (B4)
implies that there exist constants $\gamma_1, \ \gamma_2>0$ such that
(see \cite{Rab})
\begin{equation}\label{growth:F}
  F(x, \xi ) \geq \gamma_1   |  \xi   |^\mu - \gamma_2 \quad
    \forall x \in \Omega, \;  \forall \xi \in \mathbb{R}^1,
\end{equation}
it is easy to see that there exists $\phi_0\in C_0^\infty(\Omega)$ 
independent of $k\in\mathbb{N}$ such that
\[
I_k(\phi_0) \leq 0 = I_k(0).
\]
Note that due to mountain pass lemma \cite{Rab},
there exists a critical value of $I_k$ characterized by
\[
 c_k = \inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I_k(\gamma(t)),
\]
where
$\Gamma= \{   \gamma\in C([0,1];H^1(\Omega))   ;
              \gamma(0)=0, \  \gamma(1)=\phi_0   \}$.
Take $\gamma(t)=t   \phi_0$ as a test path
in $\Gamma$ for all $k\in\mathbb{N}$,
then we obtain
\begin{equation}\label{Mono37-eq}
 c_k \leq \max_{t\in[0,1]} I_k(t\phi_0) = \max_{t\in[0,1]} I_0(t\phi_0) =: c^*,
\end{equation}
which implies the boundedness of $\{c_k\}_{k=1}^{\infty}$.
Moreover, from \eqref{Monomu-eq}
it is clear that for $k\in\mathbb{N}$,
\begin{equation*}
 c_k \geq \mu   \rho^2.
\end{equation*}
Therefore, the critical point with the critical value $c_k$
gives a nontrivial solution of \eqref{Mono36-eq}.
Let $u_k$ be a critical point of $I_k$ with the critical value $c_k$,
then by using \eqref{Mono37-eq},
we can derive the $H^1$-boundedness of $\{u_k\}_{k=1}^{\infty}$.
In fact, taking $u=u_k$ in \eqref{Monomi-eq},
then from $\nabla I_k(u_k)=0$,
we have
\begin{equation}\label{Mono39-eq}
 \big(\frac{1}{2}-\frac{1}{\mu}\big)
 \int_{\Omega}(   |\nabla u_k|^2 + b   u_k^2  )   dx
\leq
 I_k(u_k) + C_\Omega \leq c^* + C_\Omega.
\end{equation}
Furthermore, we can derive the following $L^{\infty}$-estimates for $u_k$.

\begin{lemma}\label{infty-lem}
Let $n\geq2$. Then there exist $c = c  (n,p,g)>0$ and
$\gamma=\gamma   (n,p)\geq1$ such that any weak solution $u_k\in H^1(\Omega)$ of
\eqref{Mono36-eq} with $\|u_k\|_{H^1(\Omega)}\leq K$
satisfies
\[
 \|u_k\|_{L^{\infty}(\Omega)} \leq c   K^{  \gamma}.
\]
\end{lemma}

\begin{proof}
Our proof is based on Moser's iteration argument; see 
 \cite[Lemma 3.1]{O0} or \cite[Theorem 8.17]{Gilbarg-T}.
Here we use the notation $\|u\|_r=\|u\|_{L^r(\Omega)}$ for $r\in[1,\infty]$.
From (A6), we can choose $R_0>0$ such that $(\beta(u)-g(u))u\geq0$ for $|u|\geq R_0$.
We set $w_k=\max \{u_k,0 \}$ and $m_0=\sup_{|u|\leq R_0}|g(u)|/|u|$.
Then we see that
\[
 (   \beta_k(u_k)-g_k(u_k)   )   w_k\geq-m_0   w_k^2.
\]
Hence multiplying \eqref{Mono36-eq} by $w^{\alpha}$ ($\alpha\geq1$),
 we obtain
\[
 \min ( \frac{4\alpha}{(\alpha+1)^2}, b)
     \| w_k^{(\alpha+1)/2} \|_{H^1(\Omega)}^2
\leq  \int_{\Omega}w_k^\alpha F(x,u_k)dx+m_0\int_{\partial\Omega}w_k^{\alpha+1}d\sigma.
\]
From (B2) and (B3), there exists $c>0$ such that $|F(x,u)|\leq c   (|u|+|u|^{p})$.
Therefore,
\begin{equation}\label{est:H1a}
   \min ( \frac{4\alpha}{(\alpha+1)^2}, b)
       \| w_k^{(\alpha+1)/2} \|_{H^1(\Omega)}^2
\leq  c\int_{\Omega}\left( w_k^{\alpha+1}+w_k^{\alpha+p} \right)dx
 +m_0\int_{\partial\Omega}w_k^{\alpha+1}d\sigma.
\end{equation}
Furthermore, from the trace inequality, we obtain
\begin{equation}\label{est:H1b}
   \begin{split}
\int_{\partial\Omega}w_k^{\alpha+1}   d\sigma
 &=  \| w^{(\alpha+1)/2} \|_{L^2(\partial\Omega)}^2 \\
 & \leq  \big(\frac{\varepsilon}{\alpha+1}\big)
 \| \nabla w_k^{(\alpha+1)/2} \|_2^2
 +  c   \big(\frac{\alpha+1}{\varepsilon}\big)
 \| w_k^{(\alpha+1)/2} \|_2^2.
 \end{split}
 \end{equation}
Let $p^*=2n/(n-2)$ if $n\geq3$  and $p^*=2   p$ if $n=2$.
We choose $\theta\in(0,1)$ such that $1/2 p=\theta/2+(1-\theta)/p^*$ and set
$\kappa=p^*/2>1$.
By the H$\ddot{\rm o}$lder inequality and the interpolation inequality,
we have
\[
  \int_{\Omega}w_k^{\alpha +p}dx
\leq  \|w_k^{p-1}\|_{p/(p-1)}\|w_k^{\alpha+1}\|_{p}
\leq  \|w_k\|_{p}^{p-1} \| w_k^{\alpha+1} \|_{\kappa}^{1-\theta}
  \| w_k^{\alpha+1} \|_1^{\theta}.
\]
By the Sobolev inequality and the assumption $\|u_k\|_{H^1(\Omega)}\leq K$,
we see that
\begin{equation}  \label{est:H1c}
\begin{split}
&\int_{\Omega}w_k^{\alpha +p}dx\\
&\leq  c   K^{p-1} \| w_k^{\alpha+1} \|_{\kappa}^{1-\theta}
            \| w_k^{\alpha+1} \|_1^{\theta}  \\
&\leq  c\Big( \big(\frac{\varepsilon}{\alpha+1}\big)
   \| w_k^{\alpha+1} \|_{\kappa}
   + K^{(p-1)/\theta} \big(\frac{\varepsilon}{\alpha+1}\big)^{-(1-\theta)/\theta}
          \| w_k^{\alpha +1} \|_1 \Big)
\\
&\leq  c \Big( \big(\frac{\varepsilon}{\alpha+1}\big)
 \| w_k^{(\alpha+1)/2} \|_{2^*}^2
 + K^{(p-1)/\theta}\big(\frac{\varepsilon}{\alpha+1}\big)^{-(1-\theta)/\theta}
 \|w_k\|_{\alpha+1}^{\alpha +1} \Big)
\\
&\leq  c \Big( \big(\frac{\varepsilon}{\alpha+1}\big)
 \| w_k^{(\alpha+1)/2} \|_{H^1(\Omega)}^2
 + K^{(p-1)/\theta}\big(\frac{\varepsilon}{\alpha+1}\big)^{-(1-\theta)/\theta}
 \|w_k\|_{\alpha+1}^{\alpha +1} \Big).
\end{split}
\end{equation}
Hence, in view of \eqref{est:H1a}, \eqref{est:H1b} and \eqref{est:H1c},
taking $\varepsilon>0$ small enough,
we obtain
\[
  \| w_k^{(\alpha+1)/2} \|_{H^1(\Omega)}^2
\leq
 cK^{(p-1)/\theta}(\alpha+1)^{\nu}\|w_k\|_{\alpha+1}^{\alpha+1}
\]
for some $\nu\geq1$.
Therefore, from the Sobolev inequality, it follows that
\[
 \|w_k\|_{\kappa(\alpha+1)}^{\alpha+1} \leq
 cK^{(p-1)/\theta}(\alpha+1)^{\nu}\|w_k\|_{\alpha+1}^{\alpha+1}.
\]
By the same iteration argument as in the proof in \cite[Theorem 8.17]{Gilbarg-T},
we can show that here exists $\gamma>0$ such that
\[
 \|w_k\|_\infty \leq cK^{\gamma}\|w_k\|_2,
\]
which assures the $L^{\infty}$-estimates for $w_k=\max\{u_k,0\}$.
By the arguments similar to those above,
we can also derive the $L^{\infty}$-estimates for $\min\{u_k,0\}$.
Thus the proof is completed.
\end{proof}


For the case $n\geq2$, Lemma \ref{infty-lem} and \eqref{Mono39-eq}
assure that $\{u_k\}_{k=1}^{\infty}$ is bounded in $L^{\infty}(\Omega)$.
As for the case $n=1$, \eqref{Mono39-eq} with the embedding
 $H^1(\Omega)\subset L^{\infty}(\Omega)$
assures the boundedness of $\{u_k\}_{k=1}^{\infty}$ in $L^{\infty}(\Omega)$.
Hence there exists $A>0$ such that $\|u_k\|_{L^\infty(\Omega)}\leq A$.
Therefore if solutions $u_k $ are classical ones, then it is clear that
by the construction of $\beta_k$ and $g_k$, we find that
\[
 \beta_k(u_k)=\beta(u_k),\quad  g_k(u_k)=g(u_k) \quad
   \text{for } k > A.
\]
Thus it is easy to see that $u_k$ with $k>A$ satisfies
\eqref{def:weak} and gives the desired solution.
To verify this rigorously for $u_k \in H^1(\Omega) \cap L^\infty(\Omega)$,
we prepare the following lemma, which completes the proof of
Theorem \ref{exist-thm}.
\end{proof}

\begin{lemma}\label{infty-trace}
Let $\Omega$ be a domain in $\mathbb{R}^n$ such that the trace theorem
in $L^p(\Omega)$ holds true for some $p \in [1,\infty)$
and let $u$ belong to $W^{1,p}(\Omega) \cap L^\infty(\Omega)$.
Then $u$ belongs to $L^\infty(\partial \Omega)$
and satisfies
\begin{equation*}
\|  u   \|_{L^\infty(\partial \Omega)}  \le \|  u   \|_{L^\infty(\Omega)}.
\end{equation*}
\end{lemma}

\begin{proof}
 We are going to apply the ``$L^\infty$-energy method" developed in \cite{O1,O2}.
Since $u \in W^{1,p}(\Omega) \cap L^\infty(\Omega)$
assures that $| u |^{r} \in W^{1,p}(\Omega) $ 
for all $r \in[1, \infty)$ and the trace theorem works in
$L^p(\Omega)$, we have
\begin{align*}
\|   u   \|_{L^{p   s}(\partial \Omega)}
 & = \|  |u|^s   \|_{L^{p}(\partial \Omega)}^{1/s}\\
&\le \Big( C_p   \Big\{ \Big( \int_{\Omega} s^p   |u|^{(s-1)   p}
                              | \nabla u|^p   dx \Big)^{1/p}
             + \Big( \int_{\Omega} (|u|^s)^p   dx \Big)^{1/p} \Big\}
             \Big)^{1/s}
 \\
  & \le C_p^{1/s}  s^{1/s}  \|   u   \|_{L^\infty(\Omega)}^{\frac{s-1}{s}}
         \big(   \| \nabla u \|_{L^p(\Omega)} + \| u \|_{L^p(\Omega)}
            \big)^{1/s} \to \|   u   \|_{L^\infty(\Omega)}\,,
\end{align*}
as $s \to \infty$.
 Then the conclusion follows from \cite[Lemma 2.2]{O2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{many-thm}]
We again consider approximation problems
\eqref{Mono36-eq}. Since
$g(u)$ and $f(\cdot,u)$ are assumed to be odd functions,
we can apply the symmetric mountain pass lemma
to obtain infinitely many solutions for approximation problems.
In fact,
let $ < \lambda_1 < \lambda_2 \le \lambda_3 \le \cdots$ be the eigenvalues of
$- \Delta$ with homogeneous Dirichlet boundary condition and
let $e_i$ be the corresponding $i$-th eigenfunctions.
 We claim that for a sufficiently large $k_0 \in \mathbb{N}$,
there exist  $\rho>0, \ \alpha>0$ such that
$I_k(u) \ge \alpha$ for all
$u \in V^+ :=$ span $\{  e_k   ;   k \ge k_0   \}$
with $\|u \|_{H^1(\Omega)} = \rho$.
 Indeed, by (B2) and the interpolation inequality, we obtain
\begin{align*}
I_k(u)  
& \ge  \int_{\Omega} \big(  \frac{1}{2}   |\nabla u|^2 + b  |u|^2 \big)   dx
        - C \int_{\Omega}  |u|^{p}   dx          - C
   \\
& \ge  {\tilde b}   \|  u \|_{H^1}^2
             - C    \| u \|_{L^2}^r    \| u \|_{L^{2^*}}^{p-r} - C
   \\
& \ge \left( {\tilde b}
          - C_1   \lambda_{k_0}^{-r/2}  \| u \|_{H^1}^{p-2}
            \right)  \| u \|_{H^1}^{2} - C_2,
\end{align*}
where ${\tilde b} = \min ( 1/2, b)$, 
$\frac{r}{2} + \frac{p-r}{2^*} =1$. Then since $r = n   ( 1-p/2^*)>0$,
taking $\rho = \sqrt{2(C_2+1)/{\tilde b}}$ and choose $k_0 \in \mathbb{N}$ such that
$C_1   \lambda_{k_0}^{-r/2}  \rho^{  p-2} \le {\tilde b}/2$, we find that
$I_k(u) \ge 1$ for all $u \in V^+$ with$ \| u \|_{H^1} = \rho$.

 Now we put $V^- :=$ span $\{  e_k   ;   k < k_0   \}$,
 the orthogonal complement of $V^+$ in $H^1_0(\Omega)$.
Since $I_k|_{V^-} = I_0|_{V^-}$ and $V^-$ is finite dimensional,
by virtue of \eqref{growth:F}, there exists $R>0$ independent of $k \in \mathbb{N}$
such that
\[
 I_k(u) \leq 0 \quad  \text{for } u \in V^- \setminus B_{V^-}(R),
\]
where $B_{V^-}(R)=\{   u \in V^-   ;   \|u\|_{H^1(\Omega)}<R   \}$.
Then by the symmetric mountain pass lemma
\cite[Theorem 9.12]{Rab},
there exist infinitely many critical points
$\{u_k^j\}_{j=1}^{\infty}$ of $I_k$
whose critical values
$\{c_k^j\}_{j=1}^{\infty}$
are unbounded and characterized by
\[
 c_k^j = I_k(u_k^j) = \inf_{h \in \Gamma}\max_{u\in V^-}I_k(h(u)),
\]
where
$\Gamma = \{   h \in C(V^-   ;   H^1(\Omega))  ;  h  \text{ is odd}, \ h(u) = u$
                if $u \in V^- \setminus B_{V^-}(R)   \}$.
Take ${\rm id} \in \Gamma$ as a test path,
then we obtain
\[
 c_k^j \leq \max_{u\in V^-}I_k(u) = \max_{u\in V^-}I_0(u) =: c_j^*,
\]
whence follows the boundedness of $\{c_k^j\}_{k=1}^{\infty}$.
Hence the
$H^1$-boundedness of $\{u_k^j\}_{k\in\mathbb{N}}$ follows from \eqref{Mono39-eq}; i.e.,
there exists $C$ such that
\[
 \|u_k^j\|_{H^1(\Omega)}^2 \leq C   \left( I_k(u_k^j) +1 \right) \leq
 C   ( c_j^*+1).
\]
Moreover from Lemma \ref{infty-lem},
we obtain the $L^{\infty}$-estimates for $\{u_k^j\}_{k\in\mathbb{N}}$.
\[
 \|u_k^j\|_{L^{\infty}(\Omega)}
\leq
 c( 1+c_j^*)^{\gamma/2} =: A_j.
\]
Hence by the construction of $\beta_k$ and $g_k$ and Lemma \ref{infty-trace},
we note that for $k>A_j$,
\[
 \beta_k(u_k^j) = \beta(u_k^j), \quad  g_k(u_k^j) = g(u_k^j).
\]
Therefore, the critical point $u_k^j$ of the functional $I_k$ with $k\geq A_j$ 
turn out to be the critical points of $I$.
Thus, we can find infinitely many critical points $\{u_k\}_{k=1}^{\infty}$ of $I$
whose critical values are unbounded, which completes the proof.
\end{proof}

\begin{remark} \label{rmk2.3} \rm
(1) If we assume that $g$ is a locally Lipschitz continuous function on $\mathbb{R}$,
then every solution in $H^1(\Omega)\cap L^{\infty}(\Omega)$ of
\eqref{Mono5-eq} given in Theorems \ref{exist-thm} and \ref{many-thm}
belongs to $H^2(\Omega)$. In fact, Since (A3) and (A4) are satisfied with
$\beta, \ g $ replaced by $\beta_k, \ g_k$ respectively, we can apply
Theorem \ref{th:es}.

(2)  In Theorems \ref{exist-thm} and \ref{many-thm}, if $n=1$,
then (B2) can be dropped while if $n=2$, then it suffices that 
$|  f(x,\xi)   | \le c   \exp \varphi(\xi)$,  with 
$\varphi(\xi)   \xi^{-2} \to 0$ as $|\xi| \to \infty$;
see \cite{Rab}.
\end{remark}

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\end{document}