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

\AtBeginDocument{{\noindent\small
Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
{\em Electronic Journal of Differential Equations},
Conf. 19 (2010),  pp. 197--205.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{197}
\title[\hfilneg EJDE-2010/Conf/19/\hfil Steklov spectrum and nonresonance]
{Steklov spectrum and nonresonance for elliptic equations with
nonlinear boundary conditions}

\author[N. Mavinga, M. N. Nkashama\hfil EJDE/Conf/19 \hfilneg]
{Nsoki Mavinga, Mubenga N. Nkashama}  % in alphabetical order

\address{Nsoki Mavinga \newline
Department of Mathematics, University of Rochester,
Rochester, NY 14627-0138, USA}
\email{mavinga@math.rochester.edu}

\address{Mubenga N. Nkashama \newline
 Department of Mathematics, University of Alabama at Birmingham,
Birmingham, AL 35294-1170, USA}
\email{nkashama@math.uab.edu}

\thanks{Published September 25, 2010.}
\subjclass[2000]{35J65, 35J20}
\keywords{Steklov eigenvalues; elliptic equations;
nonlinear boundary conditions; \hfill\break\indent minimax methods}

\begin{abstract}
 This article is devoted to the solvability of second order
 elliptic partial differential equations with nonlinear boundary
 conditions. We prove existence results when the nonlinearity  on the
 boundary interacts, in some sense, with the Steklov spectrum. We
 obtain nonresonance results below the first Steklov eigenvalue as
 well as between two consecutive Steklov eigenvalues. Our method of
 proof is variational and relies mainly on minimax methods in
 critical point theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

This paper concerns  existence results for second
order elliptic partial differential equations with (possibly)
nonlinear boundary conditions
\begin{equation}\label{eq1ST}
\begin{gathered}
 -\Delta u + c(x)u = 0 \quad\text{in } \Omega,\\
\frac{\partial u}{\partial \nu}=  g(x,u) \quad\text{on }
\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^n$ is a bounded domain  with boundary
$\partial \Omega$ of class $C^{0,1}$, and
 $\partial/\partial\nu :=\nu\cdot\nabla$
is the outward (unit) normal derivative on $\partial\Omega$.

Throughout this paper we shall assume that $n\ge 2$ and that the
function $c:\Omega\to\mathbb{R}$, and the nonlinearity
$g:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ satisfy the
following conditions.
\begin{itemize}

\item[(C1)] $c\in L^p(\Omega)$ with $p\ge n/2$ when $n\ge 3$
($p\ge 1$ when $n=2$), and  $c\ge 0$ a.e. on $\Omega$
with strict inequality on a set of positive measure; that is,
$\int_\Omega c(x)\, dx > 0$.

\item[(C2)]  $g\in
C(\overline{\Omega}\times\mathbb{R})$.

\item[(C3)] There exist constants $a_1,\, a_2>0$ such that
$$
|g(x,u)|\leq a_1+a_2|u|^s \quad \text{with } 0\leq s<\frac{n}{n-2}.
$$
\end{itemize}

The purpose of this paper is to study the existence of weak
solutions of problem \eqref{eq1ST} in which the nonlinearity
interacts, in some sense, with the Steklov spectrum.  By a weak
solution of \eqref{eq1ST} we mean a function $u\in H^1(\Omega)$
such that
\begin{equation}\label{weaksol}
\int\nabla u\nabla v+\int c(x)uv =\oint g(x,u)v\quad  \text{for any
$v\in H^1(\Omega)$,}
\end{equation}
 where $\int$  denotes  the (volume)
integral on $\Omega$,   $\oint$ denotes the (surface)
integral on $\partial \Omega$, and throughout this paper,
$H^1(\Omega)$ denotes the usual real Sobolev space of functions on
$\Omega$.

The nonlinear problem, \eqref{eq1ST}, has been
studied by many authors in the framework of sub and super-solutions
method. We refer e.g. to Amann \cite{HA76ST}, Mawhin and Schmitt
\cite{MS85ST}, and references therein. Restricting the domain of the
nonlinearity (through a slightly modified problem) to the sub and
super solutions interval, the methods used in that framework reduce
the problem to essentially considering bounded nonlinearities and
then using a priori estimates and fixed points or topological degree
arguments. Since it is based on (the so-called) comparison
techniques, the (ordered) sub-super solutions method does not apply
when the nonlinearities are compared with higher eigenvalues.

 In recent years much work has been devoted to the study of the solvability of
elliptic boundary value problems (with linear homogeneous bounday
condition) where the reaction nonlinearity in the differential
equation interacts with the eigenvalues of the corresponding linear
differential equation with linear homogeneous boundary condition
(resonance and nonresonance problems). For some recent results in
this direction we refer to the papers by Castro~\cite{C04ST}, de
Figueiredo and Gossez \cite{FG86ST},  Rabinowitz \cite{R86ST}, and
the bibliography therein.

Concerning Problem \eqref{eq1ST} with boundary eigenparameters,
there are some (scattered) results in the literature by several
authors. For the linear case, we mention  the work by
Steklov~\cite{Stek1902ST} who initiated the problem on a disk in
1902, Amann~\cite{HA76ST}, Bandle~\cite{B80ST}, and more recently
Auchmuty~\cite{GA04ST}.  To the best of our knowledge, not much has
been done for the nonlinear problem \eqref{eq1ST} in the framework
of the Steklov spectrum. A few results on a disk ($n=2$) were
obtained by Klingelh\"ofer \cite{KK68} and Cushing~\cite{Cus72ST}.
(The results in \cite{KK68} were significantly generalized to higher
dimensions in \cite{HA76ST} in the framework of sub and
super-solutions method as aforementioned.) We  also refer to
Klingelh\"ofer \cite{KK70} where monotonicity methods were used for
nonlinearities near the first eigenvalue.

In this paper, we prove existence results when the nonlinearity
involved asymptotically interacts, in some sense, with  the Steklov
spectrum.  We derive the so-called nonresonance results below the
first Steklov eigenvalue as well as between two consecutive Steklov
eigenvalues. This appears to be the first time that the boundary
nonlinearity $g$ is compared with \emph{higher} Steklov eigenvalues.
Our method of proof is variational and relies mainly on a priori
estimates and minimax methods in critical point theory.

This paper is organized as follows.  In Section 2 we have collected
some relevant preliminary results on linear Steklov eigenproblems
which are needed for our purposes. (The proofs of these auxiliary
results may be found in a recent paper of Auchmuty \cite{GA04ST}.)
We also state our main results  which consist of relating the
asymptotic behavior of the nonlinearity involved with the Steklov
spectrum. In Section 3 we first provide  some auxiliary results on
Critical Point Theory that are needed for the proofs of our main
results. Then we give the proofs of our main results, and a few
remarks to relate our results to the previous ones in the
literature.

Unlike some previous
approaches to problems with nonlinear boundary conditions, all of
our results are based upon minimax methods in Critical Point Theory
(see e.g. Rabinowitz~\cite{R86ST} and references therein).

\section{Preliminaries on Steklov problems and main results}

To put our results into context, we have collected  some relevant
results on linear Steklov eigenproblems needed for our purposes. We
refer to a very recent and interesting paper of Auchmuty
\cite{GA04ST} for the proofs of the results regarding Steklov
eigenproblems. We then state the main results which consist of
relating, in some sense, the asymptotic behavior of the nonlinearity
involved with the first Steklov eigenvalue, then subsequently with
two consecutive higher Steklov eigenvalues.

 Consider the linear problem
\begin{equation}\label{eq2ST}
\begin{gathered}
 -\Delta u + c(x)u = 0 \quad\text{in } \Omega,\\
\frac{\partial u}{\partial \nu}=  \mu u \quad\text{on }
\partial\Omega.
\end{gathered}
\end{equation}
The Steklov eigenproblem is to find a pair
$(\mu,\varphi)\in\mathbb{R}\times H^1(\Omega)$,
$\varphi\not\equiv 0$, such that
$$
\int\nabla \varphi\nabla v+\int c(x)\varphi v =\mu\oint \varphi
v\quad \text{for any $v\in H^1(\Omega)$.}
$$
Picking $v=\varphi$, and subsequently $v\in H^1_0(\Omega)$ one
immediately sees that if there is such an eigenpair, then $\mu >0$
and $\varphi\perp H^1_0(\Omega)$
in the $H^1$-$c$-inner product defined by
\begin{equation}
(u,v)_c=\int\nabla u\nabla v+\int c(x)uv,
\end{equation}
with the associated norm denoted by $\|u\|_c$; which is
equivalent to the standard norm on $H^1(\Omega)$. This implies that
one can split
\begin{equation}\label{split}
H^1(\Omega)=H^1_0(\Omega)\oplus_c [H^1_0(\Omega)]^{\perp}
\end{equation}
as a direct orthogonal sum (in the sense of $H^1$-$c$-inner
product).

Besides the Sobolev spaces, we shall make use, in what follows, of
the real Lebesgue spaces $L^q(\partial\Omega)$, $1\leq q\leq
\infty$, and the compactness of the trace operator
$\Gamma:H^1(\Omega)\to L^q(\partial\Omega)$  for
$1\leq q<\frac{2(n-1)}{n-2}$ (see e.g. Kufner, John and
Fu\v{c}\'{\i}k~\cite[Chap.~6]{KJF77ST} and references therein).
Sometimes we will just use $u$ in place of $\Gamma u$ when
considering the trace of a function on $\partial \Omega$. Throughout
this paper we denote the $L^2(\partial \Omega)$-inner product by
\begin{equation}  (u,v)_{\partial}=\oint uv
\end{equation}
and the associated norm  by $\|u\|_{\partial}$.

Assuming that the above assumptions are satisfied, Auchmuty
\cite{GA04ST} recently proved that, for $n\ge2$, the Steklov
eigenproblem \eqref{eq2ST} has a sequence of real eigenvalues
$$
0<\mu_1\le\mu_2\le\ldots\le\mu_j\le \ldots \to\infty, \quad
\text{as $j\to\infty$,}
$$
each eigenvalue has
a finite-dimensional eigenspace. The  eigenfunctions $\varphi_j$
corresponding to these eigenvalues form a complete orthonormal
family in $[H^1_0(\Omega)]^\perp$, which is also
complete and orthogonal in $L^2(\partial \Omega)$. Moreover, the
trace inequality
\begin{equation}\label{TraceIneq}
\mu_1\oint (\Gamma u)^2\leq \int|\nabla u|^2+\int c(x)u^2
\end{equation}
holds for all $u\in H^1(\Omega)$, where $\mu_1>0$ is the least
Steklov eigenvalue for \eqref{eq2ST}. If equality holds in
\eqref{TraceIneq}, then $u$ is a multiple of an eigenfunction of
\eqref{eq2ST} corresponding to $\mu_1$.

\begin{theorem}[Below the first Steklov eigenvalue] \label{thm2ST}
 Assume {\rm (C1)--(C3)} hold. Let the potential
$ G(x,u)=\int_0^ug(x,s)\,ds$ be such that the following
condition holds.
\begin{itemize}
\item[(C4)] There exists $\mu\in\mathbb{R}$  such that
$$
\limsup_{|u|\to\infty}\frac{2G(x,u)}{u^2}\leq\mu<\mu_1
$$
uniformly for $x\in\overline{\Omega}$.
\end{itemize}
 Then the nonlinear equation \eqref{eq1ST} has at least one solution
$u\in [H^1_0(\Omega)]^\perp$.
\end{theorem}

Consequently, we derive  nonresonance below the first eigenvalue
associated with the Steklov problem. Notice that we impose
conditions on the potential of the boundary nonlinearity $g$ rather
than on $g$ itself as was done in the previous papers \cite{HA76ST,
Cus72ST, KK68, KK70}.

In the next result, we are concerned with the case where the
asymptotic behavior of the nonlinearity is related to two
consecutive Steklov eigenvalues. We impose conditions on the
asymptotic behavior of the nonlinearity $g(x,u)$  directly. These
conditions imply similar ones on the asymptotic behavior of the
potential $G(x,u)$.

\begin{theorem}[Between consecutive Steklov eigenvalues] \label{thm3ST}
Assume {\rm (C1)--(C3)} are met,
and  that the following condition holds.
\begin{itemize}
\item[(C5)] There exist constants $a, b\in\mathbb{R}$ such that
$$
\mu_j< a\le \liminf_{|u|\to\infty}\frac{g(x,u)}{u}\le
\limsup_{|u|\to\infty}\frac{g(x,u)}{u}\le b<\mu_{j+1},
$$
uniformly for $x\in\overline\Omega$.
\end{itemize}
 Then the nonlinear equation
\eqref{eq1ST} has at least one solution $u\in
[H^1_0(\Omega)]^\perp$.
\end{theorem}

We will use a variational approach to prove Theorems
\ref{thm2ST}--\ref{thm3ST}.

\section{Proofs of main results}

 We first state some auxiliary results which will be needed in
the sequel.  The following result on the continuity of the
Nemytsk\v{\i}i operator on the boundary readily follows from the
arguments similar to those used in the proof of
\cite[Proposition B.1]{R86ST}. For the last two auxiliary results,
we refer to \cite{Mav08} for the proofs.

\begin{proposition}\label{NemytskiiST}
Suppose that $g$ satisfies {\rm (C2)} and there are constants
$p,q\geq 1$  and  $a_1, a_2$ such that for all $x\in
\overline{\Omega}, \xi\in \mathbb{R}$,
$$
|g(x,\xi)|\leq a_1+a_2|\xi|^{p/q},
$$
Then the Nemytsk\v{\i}i operator $\varphi(x)\to g(x, \varphi(x))$ is
continuous from $L^p(\partial\Omega)$ to $L^q(\partial\Omega)$.
\end{proposition}

\begin{proposition}\label{functional}
Assume that {\rm (C1)--(C3)} hold. Then the functional associated with
\eqref{eq1ST} defined by $I:H^1(\Omega)\to\mathbb{R}$  with
\begin{equation}
I(u)= \frac12\Big[\int|\nabla u|^2+\int c(x)u^2\Big]-\oint G(x,u),
\end{equation}
is well-defined and of class $C^1$ with Fr\'{e}chet derivative
$I'(u)$ in $(H^1(\Omega))^*$ given by
\begin{equation}
 I'(u)v= \int\nabla u\nabla v+\int c(x)uv-\oint g(x,u)v\quad
 \text{ for every }v\in  H^1(\Omega).
 \end{equation}
Moreover,
$J(u)=\oint G(x,u)$ is weakly continuous, and
$J'$ is compact.
\end{proposition}


The next result concerns the Palais-Smale condition (PS) which
builds some ``compactness" into the functional $I$. It requires
that any sequence $\{u_m\}$ in $H^1(\Omega)$ such that (i)
$\{I(u_m)\}$ is bounded, (ii) $\lim_{m \to
\infty}I'(u_m)=0$, be precompact.

Owing to the next proposition, to get (PS) it suffices to show that
(i)--(ii) imply that $\{u_m\}$ is a bounded sequence.

\begin{proposition}\label{propPS}
Assume that {\rm (C1)--(C3)} hold. If $\{u_m\}$ is a
bounded sequence in $H^1(\Omega)$ such that
$\lim_{m\to \infty}I'(u_m)=0$, then $\{u_m\}$
 has a convergent subsequence.
\end{proposition}



Now we prove Theorems \ref{thm2ST}--\ref{thm3ST}. We will
 use of the Saddle Point Theorem and its variant proved in
\cite{R86ST}.


\begin{proof}[Proof of Theorem \ref{thm2ST}]
 Observe that condition (C4) implies that for every
$\epsilon>0$ there is $r=r(\epsilon)>0$ such that
\begin{equation}\label{cond1ST}
\frac{2G(x,u)}{u^2}\leq \mu +\epsilon
\end{equation}
for all $x\in\overline{\Omega} $ and all $u\in \mathbb{R}$ with
$|u|>r$. Combining \eqref{cond1ST} and (C3), there exists a
constant $M_\epsilon >0$ such that
\begin{equation}\label{cond2ST}
\forall x\in\overline{\Omega},\;\forall u\in \mathbb{R},\;
G(x,u)\leq \frac{1}{2}(\mu +\epsilon){u^2}+ M_\epsilon.
\end{equation}
To prove that \eqref{eq1ST} has at least one solution, it
suffices, according to \cite[Theorem 2.7]{R86ST}, to show
that the functional $I$ is bounded below and that it satisfies the
 (PS) condition. Under the assumptions of Theorem
\ref{thm2ST}, we shall  show that the functional $I$ is coercive on
$H^1(\Omega);$ that is,
\begin{equation}\label{coercivity}I(u)\to
\infty \quad \text{as } \|u\|_c\to \infty,
\end{equation}
which would imply that $I$ is bounded below and that the
Palais-Smale is satisfied.

Now let us prove that $I$ is coercive on $H^1(\Omega)$. Assume
$\|u\|_c\to \infty$, then by using the continuity of the trace
operator from $H^1(\Omega)$ into $L^{2}(\partial \Omega)$, we get
that either $\|u\|_{\partial}\to \infty$ or
$\|u\|_{\partial}<K$, where $K$ is  a positive constant.
 We claim that in either case $I(u)\to \infty $.

 First, suppose that $\|u\|_{\partial}<K$. Since
$ I(u) =\frac12\|u\|_c^2-\oint G(x,u)$, using  \eqref{cond2ST},
one has that
$ I(u)\geq
 \frac12\|u\|_c^2-\frac{1}{2}(\mu
+\epsilon)\|u\|_{\partial}^2-C$, where $C$ is a positive
constant. Hence $I(u)\to \infty $ as $\|u\|_c\to \infty$.

 Now, suppose  $\|u\|_{\partial}\to \infty$.
Using  \eqref{TraceIneq},  one has
$ I(u) \geq \frac12\left(\mu_1-(\mu +\epsilon) \right)
\|u\|_{\partial}^2-C$. Since $\mu_1>\mu$, one gets that
 $I(u)\to \infty $ as $\|u\|_c\to \infty$. Thus $I$ is coercive.

By combining condition (C3) and the coercivity of $I$, one
deduces that $I$ is bounded from below; that is,
there exists $K\in\mathbb{R}$ such  that $I(u)\geq K$,
 for all $u\in H^1(\Omega)$. It follows immediately from the
coercivity of $I$ in \eqref{coercivity} and Proposition \ref{propPS}
that $I$ satisfies (PS).   By \cite[Theorem 2.7]{R86ST},
it follows that $I$ has a critical point $u\in H^1(\Omega)$, that
is, $I'(u)=0$. Hence, $u$ satisfies \eqref{weaksol},  and thus
\eqref{eq1ST} has at least one solution. Note that
$u\in [H^1_0(\Omega)]^\perp$ by the definition \eqref{weaksol}.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3ST}]
 Under the assumptions of
Theorem \ref{thm3ST} we need to show that the conditions of the
Saddle Point Theorem \cite{R86ST} are fulfilled. Let
\begin{equation}\label{spacessplit}
V=\operatorname{span}\{\varphi_k|\, k\le j\},\quad
X= Y\oplus_c H^1_0(\Omega), \text{ where }
Y=\overline{\operatorname{span}\{\varphi_k:k\ge j+1\}}.
\end{equation}
It follows from \eqref{split}  and \eqref{spacessplit} that
\begin{equation}\label{H1split}
H^1(\Omega)=V\oplus_c X.
\end{equation}
We need to prove that there exists a
constant $r>0$ such that
\begin{equation}\label{SaddlepointCond}
 \sup_{\partial D}I<\inf_X I,
\end{equation}
where $D=\{ v\in V: \|u\|_c\leq r\}$.
Assuming that this is the case, and the Palais-Smale condition is
satisfied, we deduce by  the Saddle Point Theorem \cite{R86ST} that
$I$ has a critical point. Therefore, \eqref{eq1ST} has at least
one solution.

We shall show that the functional $I$ is coercive on $X$ and $-I$ is
coercive on $V$  which would imply that \eqref{SaddlepointCond} is
satisfied by choosing $r>0$ sufficiently large.

Notice that condition (C5) implies a similar condition on the
potential $G$; that is, the constants $a, b
\in\mathbb{R}$ are such that for all $x\in\overline{\Omega}$,
\begin{equation}\label{nonrescondG}
a\le \liminf_{|u|\to\infty}\frac{2G(x,u)}{u^2}\le
\limsup_{|u|\to\infty}\frac{2G(x,u)}{u^2}\le b.
\end{equation}
Combining (C3) and \eqref{nonrescondG}, one gets that
for every $\epsilon>0$, all $x\in\overline{\Omega}$, all
$u\in \mathbb{R}$, we have
\begin{equation}\label{nonrescond}
(a-\epsilon)\frac{u^2}{2}-C \leq G(x,u)\leq
(b+\epsilon)\frac{u^2}{2}+ C\ ,
\end{equation}
where $C=C(\epsilon)$ is a positive constant.

On the one hand,  it follows that for every $u \in V$ one has that
$$
I(u)=  \frac12\|u\|_c^2 -\oint G(x,u) \leq\frac12\|u\|_c^2-
\frac{1}{2}(a-\epsilon)\oint u^2 +\tilde{C}
= \frac{1}{2}\|u\|_c^2-\frac{1}{2}(a-\epsilon)\|u\|^2_\partial +\tilde{C}.
$$
Using the Parseval identities obtained in \cite[p. 331]{GA04ST} it
follows that
\[
 I(u)\leq \frac12\Big(1-\frac{a}{\mu_j}+\frac{\epsilon}{\mu_j}\Big)
\|u\|_c^2 +\tilde{C}.
\]
Since $\mu_j<a$, it follows that
$1-\frac{a}{\mu_j}+\frac{\epsilon}{\mu_j}<0$, provided
$\epsilon>0$ is sufficiently small. Therefore, by going to the limit
as $\|u\|_c \to \infty$, one gets
\begin{equation}\label{coercV}
I(u) \to -\infty.
\end{equation}

On the other hand, for every $u\in X$, it follows from
\eqref{spacessplit} that $u= u^0+ \overline{u}$, where $u^0\in
H^1_0(\Omega) $ and $\overline{u}\in Y$. Taking into account the
$c$-orthogonality of $\overline{u}$ and $u^0$ in $H^1(\Omega)$,  one
has
$$
I(u)=\frac12\|u^0\|_c^2+\frac12\|\overline{u}\|_c^2
-\oint G(x,u) \ge \frac12\|u^0\|_c^2+\frac12\left(
\|\overline{u}\|_c^2- (b+\epsilon)\|\overline{u}\|^2_\partial
\right)-\tilde{C}.
$$
 Therefore, using the Parseval identities
obtained in \cite[p. 331]{GA04ST} it follows that
\[
I(u)\geq \frac12\|u^0\|_c^2+\frac12\Big(1-
\frac{b}{\mu_{j+1}}-\frac{\epsilon}{\mu_{j+1}} \Big)
\|\overline{u}\|_c^2-\tilde{C}.
\]
Since $b<\mu_{j+1}$, it follows that for $\epsilon>0$ sufficiently
small, $ 1-\frac{b}{\mu_{j+1}}-\frac{\epsilon}{\mu_{j+1}}>0$.
Therefore,
$$
I(u)\geq \frac12\Big(1- \frac{b}{\mu_{j+1}}-\frac{\epsilon}{\mu_{j+1}}
\Big)\|{u}\|_c^2-\tilde{C}.
$$
By going to the limit as $\|u\|_c \to \infty$, on gets
$$
I(u)\to \infty\quad\text{as $\|u\|_c\to\infty$}.
$$
Thus, $I$ is coercive on $X$. Furthermore,
it follows from the coercivity of $I$ on $X$  and condition
(C3) that $I$ is bounded below by a constant on $X$. Therefore,
using \eqref{coercV} we obtain the assertion \eqref{SaddlepointCond}
for some constant $r>0$.

It remains to prove that the functional $I$ satisfies the
Palais-Smale condition. It suffices, according to Proposition
\ref{propPS},  to show that for any sequence $\{u_m\}$ in
$H^1(\Omega)$  such that $\{I(u_m)\}$ is bounded and
$\lim_{m \to \infty}I'(u_m)=0$, it follows that
$\{u_m\}$ is bounded.

Notice that Condition (C5) implies that for every $\epsilon>0$
there exists $r>0$ such that for $|u|\ge r$,
\begin{equation}\label{ineqg}
a-\epsilon\leq \frac{g(x,u)}{u}\leq b+\epsilon \quad
 \text{for all } x\in \overline{\Omega}.
\end{equation}
 Let us define $\gamma:
\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$ by
\[
\gamma(x,u)=\begin{cases}
\frac{g(x,u)}{u}&\text{for }|u|\ge r\\[4pt]
\frac{g(x,r)+g(x,-r)}{2r^2}u+\frac{g(x,r)
-g(x,-r)}{2r}&\text{for }|u|<r.
\end{cases}
\]
The function $\gamma$ is continuous in $\overline{\Omega}\times
\mathbb{R}$ since $g$ is,  moreover by \eqref{ineqg} one has
\begin{equation}\label{ineqgamma}
a-\epsilon\leq \gamma(x,u)\leq b+\epsilon\quad
\text{for all $u\in\mathbb{R}$ and all } x\in \overline{\Omega}.
\end{equation}
Define $h:\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$  by
\begin{equation}\label{defh}
h(x,u)= g(x,u)-\gamma(x,u) u,
\end{equation}
then it follows from  the continuity of $g$ and $\gamma$ that
\begin{equation}\label{Boundh}
|h(x,u)|\leq K,
\end{equation}
 for all $(x,u)\in \overline{\Omega}\times
\mathbb{R}$, where $K>0$ is a constant.


Now, let $\{u_m\}\subset H^1(\Omega)$ be
 such that $\{I(u_m)\}$ is bounded and $\lim_{m \to
\infty}I'(u_m)=0$.  By \eqref{H1split} one has $u_m= v_m+ x_m$,
where $v_m\in V$ and $x_m\in X$. Moreover, by \eqref{spacessplit}
$x_m=x_m^0 +\overline{x}_m$, where $x_m^0\in H^1_0(\Omega)$ and
$\overline{x}_m \in Y.$\\
Since $\lim_{m \to \infty}I'(u_m)=0$, it follows that
for every $\epsilon>0$, there exists $N>0$ such that for all $m\geq
N$,
$$
\sup_{\varphi\neq 0}\frac{|I'(u_m)\varphi|}{\|\varphi\|_c}<\epsilon.
$$
Set  $\varphi=x_m-v_m $ for $m$ large.
Then, $I'(u_m)(x_m-v_m)< \epsilon\|x_m-v_m\|_c$. Taking into
account the $c$-orthogonality of $x_m$ and $v_m$ in $H^1(\Omega)$ and
\eqref{defh}, one gets from the definition of $I'$ that
\begin{align*}
&\|x_m\|_c^2-\|v_m\|_c^2 -\oint
\gamma(x,u_m)\overline{x}_m^2+\oint\gamma(x,u_m)v_m^2\\
 &<\epsilon
(\|x_m\|_c+\|v_m\|_c)+\oint h(x,u_m)\overline{x}_m-\oint
h(x,u_m)v_m.
\end{align*}
By using \eqref{ineqgamma}, \eqref{Boundh} and the continuity of the
trace operator, one obtains
\begin{align*}
&\left(\|x^0_m\|_c^2 +\|\overline{x}_m\|_c^2\right)
-\|v_m\|_c^2-(b+\epsilon)\|\overline{x}_m\|_{\partial}^2
+(a-\epsilon)\|v_m\|_{\partial}^2\\
&<\epsilon(\|x_m\|_c+\|v_m\|_c)+\widetilde{K}\|\overline{x}_m\|_c
+\widetilde{K}\|v_m\|_c,
\end{align*}

Now, using the Parseval identities obtained in \cite[p. 331]{GA04ST}
it follows that
\begin{align*}
&\|x^0_m\|_c^2+\big(1-\frac{b}{\mu_{j+1}}-\frac{\epsilon}{\mu_{j+1}}
\big)\|\overline{x}_m\|_c^2
+\big(\frac{a}{\mu_j}-\frac{\epsilon}{\mu_j}-1\big)\|v_m\|_c^2\\
&<K_0(\|x_m\|_c+\|v_m\|_c).
\end{align*}
Since $b<\mu_{j+1}$ and $\mu_j<a$, one has that, for $\epsilon>0$
sufficiently small,
\[
\delta\left(\|x^0_m\|_c^2+\|\overline{x}_m\|_c^2+\|v_m\|_c^2\right)
<K_0(\|x_m\|_c+\|v_m\|_c),
\]
where $ 0<\delta<\min\big\{\big(1-\frac{b}{\mu_{j+1}}
-\frac{\epsilon}{\mu_{j+1}}\big),
\big(\frac{a}{\mu_j}-\frac{\epsilon}{\mu_j}-1\big) \big\}$.
Hence,
\begin{equation*}
 \|u_m\|_c^2 <\widetilde{K}_0\|u_m\|_c,
\end{equation*}
which implies that $\{u_m\}$ is bounded in $H^1(\Omega)$. Therefore,
by Proposition \ref{propPS}, $I$ satisfies the Palais-Smale
condition. The proof is complete.
\end{proof}

\begin{remark}\label{Remark1ST}
{\rm  This appears to be the first time that the boundary
nonlinearity $g$ is compared with higher Steklov eigenvalues (also
see e.g.~\cite{Mav08}). Even at the first Steklov eigenvalue, we
impose conditions on the potential of the boundary nonlinearity $g$
rather than on $g$ itself as was done in previous papers. Notice
that, in this case, we do not require a (one-sided) linear growth on
$g$ as was done in \cite{HA76ST, KK68, KK70} nor do we require
monotonicity conditions as was done in \cite{KK68, KK70}.
This work is still in progress, and we hope that more general
results will appear elsewhere.}
\end{remark}

\begin{remark}\label{Remark3ST}
{\rm Our results remain valid if one considers nonlinear equations
with a more general linear part (in divergence form) with variable
coefficients.
\begin{equation}\label{GeqST}
\begin{gathered}
 -\sum_{i,j=1}^n\frac{\partial}{\partial
x_j}\Big(a_{ij}(x)\frac{\partial u}{\partial x_i}\Big)
+ c(x)u = 0 \quad\text{in } \Omega,\\
\frac{\partial u}{\partial \nu}+\sigma(x)u=  g(x,u) \quad\text{on }
\partial\Omega,
\end{gathered}
\end{equation}
where $\sigma\in L^\infty(\partial\Omega)$ with $\sigma(x)\ge 0$
a.e. on $\partial\Omega$, and $\partial/\partial \nu:=\nu\cdot
A\nabla$ is the outward (unit) conormal derivative. The matrix
$A(x):=\big(a_{ij}(x)\big)$ is symmetric with $a_{ij}\in
L^\infty(\Omega)$ such that there is a constant $\gamma>0$ such that
for all $\xi\in\mathbb{R}^n$,
$$
\langle A(x)\xi,\xi\rangle\ge\gamma|\xi|^2\quad\text{a.e. on
$\Omega$}.
$$}
\end{remark}


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\end{document}