\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 $1
0$ 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 $2k, \\
|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 $1
0$ 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)}