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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 167, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/167\hfil Multiplicity of solutions]
{Multiplicity of solutions to fourth-order superlinear
elliptic problems under Navier conditions}

\author[E. D. d. Silva, T. R. Cavalcante \hfil EJDE-2017/167\hfilneg]
{Edcarlos D. da Silva, Thiago Rodrigues Cavalcante}

\address{Edcarlos Domingos da Silva (corresponding author) \newline
Universidade Federal de Goi\'as,
IME, Caixa Postal 131,
CEP: 74001-970, Goi\^ania - GO, Brazil}
\email{edcarlos@ufg.br}

\address{Thiago Rodrigues Cavalcante \newline
Universidade Federal de Goi\'as,
IME, Caixa Postal 131,
CEP: 74001-970, Goi\^ania - GO, Brazil}
\email{thiagocavalcantegyn@gmail.com}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted March 1, 2017. Published July 5, 2017.}
\subjclass[2010]{35J20, 35J25, 35J50}
\keywords{Fourth order elliptic problems; variational methods; 
\hfill\break\indent multiplicity of solutions;
superlinear elliptic problems; nonquadraticity condition}

\begin{abstract}
 We establish the existence and multiplicity of solutions for a class of
 fourth-order superlinear elliptic problems under Navier conditions
 on the boundary. Here we do not use the Ambrosetti-Rabinowitz condition;
 instead we assume that the nonlinear term is a nonlinear function which
 is nonquadratic at infinity.
\end{abstract}

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

\section{Introduction}

In this work we shall consider the fourth-order elliptic problem
 \begin{equation}\label{pi}
\begin{gathered}
 \alpha \Delta^{2} u + \beta \Delta u  = f(x,u) \quad\text{in } \Omega, \\
 u = \Delta u  = 0  \quad \text{on } \partial \Omega,
 \end{gathered}
\end{equation}
where $\Delta^{2} = \Delta \circ \Delta$ is the biharmonic operator,
 $ N \geq 4, \Omega \subset \mathbb{R}^N$ is a smooth bounded domain,
$\alpha > 0$, $\beta \in (-\infty, \alpha \lambda_{1})$.
Problem \eqref{pi} is called fourth-order elliptic problem under Navier boundary
conditions. Here and throughout this paper $\lambda_{1}$ denotes the first
 eigenvalue problem on $( -\Delta , H^{1}_{0}(\Omega))$.
The nonlinear term $f$ is a continuous function which is superlinear at
 infinity and at the origin. Latter on, we shall consider the assumptions on
the nonlinear term $f$.

Semilinear elliptic problems involving operators of fourth order have been
considered since the pioneer paper Lazer and Mckenna \cite{Mckenna3}.
In that work Lazer and Mackenna modeled nonlinear oscillations for suspensions
bridges. It is worthwhile to mention that problem \eqref{pi} models
static deflection of an elastic plate in a fluid, see
 \cite{Mackenna1,Mackenna2,Mckenna3,Mackenna4,Mackenna5}.
The same problems can be used to describe the static form change of beam or
the motion of rigid body. Equations of this type have been extensively
studied during the last years. Here we refer the reader to
\cite{Azorero,chabrowski,JVRocalli,Gazzola,JValdo,EdJVMax,Pistoia,
Pu,Yang,Ye,wang,Zhou} and references therein.
In these papers existence and multiplicity of solutions have been considered
using several assumptions on the nonlinear term $f$. Most of them considered
the case $f(x,t) = b[(t + 1)^{+} - 1]$ or $f$ satisfying the well known
 Ambrosetti-Rabinowitz superlinear condition at infinity.

The main goal in this work is to consider fourth-order elliptic problems
without the Ambrosetti-Rabinowitz condition introduced in \cite{AR}.
The main difficulty arises from the fact
that Palais-Smale sequences are not necessary bounded under our assumptions.
To overcome this difficulty we apply the nonquadraticity condition introduced
 by Costa-Magalh\~es \cite{CM} proving that any Cerami sequences are necessary
bounded, see Section 2 ahead. It is important to emphasize that compactness
results such as Cerami condition is a powerful tool in order to apply
variational methods.

Notice that fourth-order elliptic problems are modeled in the working space
$\mathcal{H} = H_0^1(\Omega)\cap H^2(\Omega) $. This space is an Hilbert space endowed with the norm
\begin{equation*}
\|u\| = \Big(\int_{\Omega} \left( \alpha |\Delta u|^{2}
 - \beta |\nabla u|^{2} \right)dx\Big)^{1/2}, \quad u \in \mathcal{H},
\end{equation*}
and the inner product
\begin{equation*}
\langle u, v \rangle = \int_{\Omega} \left( \alpha \Delta u \Delta v
- \beta \nabla u \nabla v \right)dx, \quad u , v \in \mathcal{H}.
\end{equation*}
The weak solutions for problem \eqref{pi} are precisely the critical points
for the functional of $C^{1}$ class $I: \mathcal{H} \to \mathbb{R}$
given by
\begin{equation}\label{Funcional}
I(u) = \frac{1}{2} \int_{\Omega} \left(\alpha |\Delta u|^{2}
- \beta |\nabla u|^{2} \right)\, dx - \int_{\Omega} F(x,u)\,dx,
\end{equation}
where the primitive for $f$ is denoted by $F(x,t) = \int_{0}^{t} f(x,s) ds$,
$x \in \Omega$, $t \in \mathbb{R}$. More specifically, given $u \in \mathcal{H}$
we have that $I'(u)$ belongs to $\mathcal{H}'$ and
\begin{equation*}
I'(u) v = \langle u, v \rangle - \int_{\Omega} f(x, u) v d x \quad
\text{for any }  u , v \in \mathcal{H}.
\end{equation*}
Here $I'(u) v$ is standard for the duality product between $\mathcal{H}$ and
$\mathcal{H}'$. Furthermore, $u \in \mathcal{H}$ is a critical point for
$I$ if and only if $u$ is a weak solution to the elliptic problem \eqref{pi}.

In this work we denote $(\lambda_{i})$ the sequence of eigenvalues on
$(-\Delta, H_0^1(\Omega))$. Consider the eigenvalue problem
\begin{equation}\label{pv}
\begin{gathered}
 \alpha \Delta^{2} u + \beta \Delta u = \mu u \quad \text{in } \Omega, \\
 u = \Delta u  = 0  \quad \text{on }  \partial \Omega.
 \end{gathered}
 \end{equation}
It is easy to verify that $\mu \in \mathbb{R}$ is an eigenvalue for
 problem \eqref{pv} if only if
\begin{equation*}
\mu_{i} = \lambda_{i}\left( \lambda_{i} \alpha - \beta \right), \quad i \in \mathbb{N}.
\end{equation*}
Furthermore, the eigenfunctions for the problem \eqref{pv} are the
eigenfunctions for the problem $(-\Delta, H_0^1(\Omega))$.

Throughout this work we assume that
 $f \in C^{0} (\bar{\Omega} \times \mathbb{R},\mathbb{R})$.
 Furthermore, we shall consider the following hypotheses
\begin{itemize}
 \item [(H1)] There exist $a_{1} > 0$ and $ p \in ( 2 , 2_{*})$ such that
 $$
|f(x, t)| \leq a_{1}(1 + |t|^{p-1}),\quad \text{for any } (x,t) \in \Omega \times
 \mathbb{R}
$$
 where $ 2_{*} = 2 N /(N - 4)$.

 \item [(H2)] $\lim_{|t|\to \infty} \frac{f(x,t)}{t}= + \infty$ uniformly
in $\Omega$;

 \item [(H3)] There exists $f_{0} \in [ 0, \mu_{1})$ such that
 \[
 \lim_{|t| \to 0} \frac{f(x,t)}{t} = f_{0} \quad \text{uniformly in } \Omega,
 \]
\end{itemize}
where $\mu_{1} = \lambda_{1}(\alpha\lambda_{1} - \beta) > 0$.

It is worthwhile to mention that hypothesis (H3) implies that
\begin{equation*}
f(x, 0) = 0, \quad  \forall  x \in \Omega.
\end{equation*}
As a consequence $u \equiv 0$ is a trivial solution to the elliptic
problem \eqref{pi}. Hence, applying (H1)--(H3), the main objective in the
 present work is to verify the existence of nontrivial solutions.

It is important to mention that $2_{*} = 2 N /(N - 4)$ is the critical
Sobolev exponent for fourth order elliptic equations for any $N \geq 4$.
More precisely, we have that $\mathcal{H}$ is continuous embedding into
$L^{s}(\Omega)$ for any $s \in [1,2_{*}]$ where $N \geq 5$.
For the case $N = 4$ we have that $\mathcal{H}$ is not included in
$L^{\infty}(\Omega)$ and for this case we consider
$p \in (2, 2_{*})$ in the following form $2 < p < \infty$.
Notice also that the embedding $\mathcal{H} \subset L^{s}(\Omega)$ is compact
for any $s \in [1, 2_{*})$ where we put $N \geq 4$.

Under hypotheses (H1)--(H3) problem \eqref{pi} is superlinear at infinity and
 at the origin. In \cite{Pistoia,Pu} the authors considered fourth order elliptic
problems where the nonlinear term is a function that satisfies the well
known Ambrosetti-Rabinowitz condition at infinity.
Namely, the Ambrosetti-Rabinowitz condition, in short (AR) condition, says that:
There are $\theta > 2$ and $R > 0$ such that
\begin{equation*}
0 < \theta F(x,t) \leq t f(x,t), \quad |t| \geq R,\; x \in \Omega.
\end{equation*}
Using a standard procedure recall that $(AR)$ condition implies
\begin{equation}\label{i1}
F(x,t) \geq c_{1}|t|^{\theta} - c_2, \quad t \in \mathbb{R}, \;x \in \Omega
\end{equation}
holds  for some $c_{1}, c_2 > 0$. However, there are superlinear functions
 $f$ that \eqref{i1} is not satisfied. For example  the function
$f(t) = t \ln(1 + |t|)$ which does not satisfy the estimate \eqref{i1}
proving that (AR) condition does not work anymore.
At the same time, the function $f$ satisfies the nonquadraticity condition
at infinity given by hypotheses (NQ) ahead.

The main feature in this work is to find existence and multiplicity of solutions
for fourth order elliptic problems given by problem \eqref{pi} where the
 nonlinear term is nonquadratic at the infinity. As was mentioned before
in this work is not required that $f$ satisfy the (AR) condition.
For further results on elliptic problems without the (AR) condition we infer
 the reader to \cite{Brown,Artigo ED,UIL,JJ,ZW,MS,SHE,SWZ,SZ,Wang}
 and references therein.

At this moment we shall consider the nonquadraticity condition at infinity
 introduced by Costa and Magalh\~aes \cite{CM} stated in the  form
\begin{itemize}
 \item[(NQ)] setting $H(x,t) := f(x,t)t- 2 F(x,t)$, we have that
 $$
\lim_{|t| \to \infty} H(x,t) = +\infty,~~~\text{uniformly for } x
\in \Omega.
 $$
 \end{itemize}

Now we shall consider the Mountain Pass Theorem, under the Cerami condition,
writing our main first result in the form:

\begin{theorem} \label{thm1}
Suppose that $f$ satisfies {\rm (H1)--(H3)} and {\rm (NQ)}.
Then problem \eqref{pi} admits at least one nontrivial solution.
\end{theorem}

\begin{remark} \label{rmk1.2}\rm
Notice that Theorem \ref{thm1} holds  for Dirichlet boundary conditions.
More specifically, we consider the elliptic problem
 \begin{gather*}
 \alpha \Delta^{2} u + \beta \Delta u  = f(x,u) \quad \text{in } \Omega, \\
 u = 0, \quad  \frac{ \partial u} {\partial \eta}  = 0  \quad \text{on }
 \partial \Omega,
 \end{gather*}
where $\frac{ \partial u} {\partial \eta}$ denotes the normal derivative
on the boundary. For this problem the energy functional
$I: H_{0}^{2}(\Omega) \to \mathbb{R}$ is given by
\begin{equation*}
I(u) = \frac{1}{2} \int_{\Omega} \left( \alpha |\nabla u|^{2}
- \beta |\nabla u|^{2} \right) dx
 - \int_{\Omega} F(x, u) dx, \quad  u \in H^{2}_{0}(\Omega).
\end{equation*}
Here the first eigenvalue $\mu_{1} > 0$ can be characterized by
\begin{equation*}
\mu_{1} = \inf \Big\{ \int_{\Omega} \left( \alpha |\nabla u|^{2}
- \beta |\nabla u|^{2} \right) dx, \|u\|_2 = 1, u \in H^{2}_{0}(\Omega) \Big\}.
\end{equation*}
For more details on this subject we refer the reader to \cite{gazzola,gazzola2}.
\end{remark}

\begin{remark} \label{rmk1.3}\rm
Here we assume that $N = 1,2,3$ and $p \in (1, \infty)$ where the function
 $f$ is subcritical. Using assumptions (H1)--(H3) the energy functional
$I$ admits also the mountain pass geometry. Besides that, using hypothesis
(NQ), we also mention that $I$ verifies the Cerami condition. Hence the
Theorem \ref{thm1} remains true in this setting.
\end{remark}

Now, using a truncation technique for fourth \-order elliptic problems,
taking into account the Strong Maximum Principle for elliptic equations
we can write our second result in the following form.

\begin{theorem} \label{thm2}
Suppose that $f$ satisfies {\rm (H1)--(H3)} and {\rm (NQ)}.
Then problem \eqref{pi} admits at least two nontrivial solutions
$u, w \in \mathcal{H}$ satisfying $u > 0$ and $w < 0$ in $\Omega$.
\end{theorem}

Now, using some symmetric conditions and the Symmetric Mountain Pass Theorem,
 our third result can be stated in the following form

\begin{theorem} \label{thm3}
Suppose that $f$ satisfies {\rm (H1)--(H3)} and {\rm (NQ)}.
Assume also that $t \to f(x, t)$ is an odd function for any $x \in \Omega$ fixed.
Then the problem \eqref{pi} admits infinitely many nontrivial solutions.
\end{theorem}

Fourth-order elliptic problems involving biharmonic operator have been widely
studied during the previous years, see \cite{Azorero,chabrowski,Gazzola,Pistoia,Pu}.
In most of them was employed variational methods such as the Mountain Pass
Theorem where $\alpha = 1$ and $\beta = 0$ and $f_{0} \equiv 0$.
For example in \cite{Pistoia} the authors considered  problem \eqref{pi}
under those conditions and the nonlinear $f$ appears as simple power or
it satisfies the (AR) condition. In \cite{Pu} was considered
problem \eqref{pi} with $\alpha = 1$, $\beta < \lambda_{1}$ and $f$
superlinear at infinity. Furthermore, in \cite{Pu} the authors considered the
 hypothesis
\begin{equation}\label{cm}
\liminf_{|t| \to \infty} \frac{tf(x,t) - 2 F(x,t)}{|t|^{\sigma}} \geq a ,
\quad t \in \mathbb{R},\; x \in \Omega
\end{equation}
holding uniformly in $x \in \Omega$ for some $\sigma > \frac{N}{4}(p - 1)$
 where $a > 0$ is a suitable constant. Here we mention also that hypotheses
in the spirt of \eqref{cm} was introduced by Costa-Magalh\~aes in \cite{CM}.
In other works the authors have  considered the  assumption
\begin{equation}\label{cc}
t \to \frac{f(t)}{t}  \text{ is an increasing function for each }  t > 0.
\end{equation}
The assumption \eqref{cc} is crucial in order to ensure that any
Palais-Smale sequence is bounded, see \cite{David,MS}.
For further results on this subject we refer the reader to
\cite{Badiale,JJ,M.W.}. As an example for our setting we consider
the following functions
\begin{itemize}
 \item[(a)] $f_{1}(t) = t \ln (1 + |t|) + \sin t$,  $t \in \mathbb{R}$ where
$\mu_{1} > 1$,
 \item[(b)] $f_2(t) = t \ln (1 + |t|) + a t \sin t$, $t \in \mathbb{R}$ where
$a \in (0, 1/2)$.
\end{itemize}
Here we mention that $f_{1}, f_2$ do not satisfy neither the (AR) condition
nor assumption \eqref{cc}. Indeed, these functions do  not satisfy the estimate
given in \eqref{i1}. Furthermore, due the periodic term we observe that
 $$
\frac{d}{d t} \big[\frac{f_{1}(t)}{t} \big]\quad\text{and} \quad
\frac{d}{d t} \big[\frac{f_2(t)}{t} \big]
$$
are sign changing functions.
The main point in this work is to consider the extremal case putting
$\sigma = 0$ in assertion \eqref{cm}. Moreover, we consider the case where
assumption \eqref{cc} is not verified. Hence our results extend and
complement the early results above-mentioned.

To ensure multiplicity of solutions for the elliptic problem \eqref{pi}
 we have a difficulty because $u^{+}$ does not belong to $\mathcal{H}$ in general.
This problem is inherent to elliptic operators of higher order such as
 problem \eqref{pi}. To overcome this difficulty we shall use a truncation
technique together with strong maximum principle proving that
problem \eqref{pi} admits at least a weak solutions with constant sign.
Here the main point is to consider an auxiliary elliptic problem and
an elliptic system in order to use Stampacchia's result. On the other hand,
 assuming that $f$ is an odd function, the functional $I$ is even which allow
us to apply the Symmetric Mountain Pass Theorem under the Cerami condition
given by Ambrosetti-Rabinowitz. The main strategy here is to prove that $I$
is anti-coercive in an appropriate sense, see Section 3 ahead.
Hence our results extend and complement early results above-mentioned proving
existence and multiplicity of solutions which are defined in sign.

The paper is organized as follows:
In Section 2 we give the variational framework to the elliptic problem
 \eqref{pi} proving the compactness condition, i.e, the Cerami condition.
 Section 3 is devoted to the proof of our main results. In Appendix we
prove that any critical point $u \in \mathcal{H}$ for the functional $I$
satisfies $\Delta u = 0$ on $\partial \Omega$.

Throughout this work $C, C_{1},C_2, \ldots$ denote positive constants.
The norm in $L^{p}(\Omega)$ is denoted by $\|\cdot\|_{p}$ for each
$p \in [1, \infty]$.
For any function $u \in L^{p}(\Omega)$ we write $u = u^{+} + u^{-}$
where $u^{+} = \max(u, 0)$ and $u^{-} = \min(u, 0)$, $p \in [1, \infty]$.
The norm in $\mathcal{H}$ is denoted by $\|\cdot\|$.

\section{Variational framework}

In this section we shall prove some properties related to the elliptic
problem \eqref{pi} given a variational framework for our problem.

Let $\mathcal{H}$ be a Banach space endowed with the norm $\|\|$.
Consider $I : \mathcal{H} \to \mathbb{R}$ a functional of $C^{1}$ class.
A sequence $(u_{n}) \in \mathcal{H}$ is said to be a Palais-Smale sequence
at level $c \in \mathbb{R}$, in short $(PS)_{c}$, when $I(u_{n}) \to c$ and
$\|I'(u_{n})\|_{\mathcal{H}'} \to 0$ as $n \to \infty$.
 Recall that $I$ satisfies the Palais-Smale condition at the level $c$,
in short $(PS)_{c}$ condition, when any $(PS)_{c}$ sequence admits a convergent
subsequence. We say simply that $I$ verifies the Palais-Smale condition when
$(PS)_{c}$ condition holds true for any $c \in \mathbb{R}$.
Similarly, a sequence $(u_{n}) \in \mathcal{H}$ is said to be a Cerami
sequence at the level $c \in \mathbb{R}$, in short $(Ce)_{c}$ sequence,
whenever $I(u_{n}) \to c$ and $(1 + \|u_{n}\|) \|I'(u_{n})\|_{\mathcal{H}'} \to 0$
as $n \to \infty$. The functional $I$ satisfies the Cerami condition at the level
$c \in \mathbb{R}$, in short $(Ce)_{c}$ condition, whenever any Cerami sequence
at the level $c$ possesses a convergent subsequence.
When $I$ satisfies the Cerami condition at any level $c \in \mathbb{R}$
we say purely that $I$ satisfies the Cerami condition, in short $(Ce)$ condition.
Here we refer the reader to \cite{Badiale,David,M.W.}.

Now we recall the classical mountain pass theorem given in \cite{AR}.

\begin{theorem}\label{mountain}
Let $\mathcal{H}$ be a Banach space and $I : \mathcal{H} \to \mathbb{R}$ a
functional of $C^{1}$ class. Suppose that $I$ admits the following mountain
pass geometry
\begin{itemize}
 \item[(i)] There exist $r > 0$, $ \rho > 0 $ such that $I(u)\geq \rho > 0$,
for any $u \in \mathcal{H}$ with $\|u\| = r$.
 \item[(ii)] There exists $e \in \mathcal{H}$ with $ \| e \| > r$ such that
$I(e)\leq 0$.
\end{itemize}
Suppose also that $I$ satisfies the $(PS)_{c}$ condition or $(Ce)_{c}$
condition where we define
\begin{equation}\label{nivel}
c = \inf_{\gamma \in \Gamma} \max_{t \in [0,1]} I (\gamma(t))
\end{equation}
where $\Gamma := \{ \gamma \in C^{0}([0,1], \mathcal{H}) : \gamma(0) = 0,
\gamma(1) = e\}$. Then $c$ is a critical value which satisfies $c \geq \rho$.
\end{theorem}

Now we shall prove the Cerami condition using the nonquadraticity condition
at infinity. The main difficulty here is to ensure that any Cerami sequence
is bounded.

\begin{proposition}\label{cerami}
Suppose that $f$ satisfies {\rm (H1)--(H3)}  and {\rm (NQ)}.
Then the functional $I$ satisfies the Cerami condition at any level
 $c \in \mathbb{R}$.
\end{proposition}

\begin{proof}
Let $(u_n) \subset \mathcal{H}$ be a sequence in such way that
$$
I(u_n) \to c,\quad \|I'(u_n)\|_{\mathcal{H}'}(1+\|u_n\|) \to 0,
$$
where $c \in \mathbb{R}$. Since $f$ has subcritical growth it suffices to prove that $(u_n)$ is bounded sequence.

The proof follows arguing by contradiction. Suppose that, up to a subsequence,
$\|u_n\| \to +\infty$ as $n\to+\infty$. Setting
$v_n := u_n/\|u_n\|$ we obtain $\|v_{n}\| = 1$ and there exits
$v \in \mathcal{H}$ in such way that $v_{n} \rightharpoonup v$ in $\mathcal{H}$.
In this case, along a subsequence, we obtain
\begin{equation} \label{convPS}
\begin{gathered}
v_n \rightharpoonup v \quad \text{weakly in } \mathcal{H},\\
v_n \to v \quad \text{strongly in }L^{q}(\Omega),  \\
v_n(x) \to v(x), \quad \text{a. e. in }  \Omega,  \\
|v_n(x)| \leq h_{q}(x), \quad h \in L^{q}(\mathbb{R}^N),
\end{gathered}
\end{equation}
holds  for any $2 \leq q <2_*$.

At this stage we claim that $v \neq 0$. This fact can be proved arguing
by contradiction. In fact, assuming that $v \equiv 0$, it follows from
(H1) and (H3) that
 \begin{equation}\label{Ce1}
|F(x,u)| \leq C |u|^{2} + C |u |^{p} , \quad
\forall  (x,u) \in\Omega \times \mathbb{R}.
\end{equation}
holds for some $C > 0$. Fix $ m> $ 0 a constant. Taking into account
\eqref{Ce1} we have
\begin{equation*}
\int_{\Omega} |F(x,\sqrt{4m}v_{n})|\, dx
\leq C \int_{\Omega} v_{n}^{2}\, dx + C \int_{\Omega} |v_{n}|^{p}\, dx
\end{equation*}
holds for some $C = C(\Omega, f, m, p) > 0$. Using the strong convergence
in \eqref{convPS} and the compact embedding we know that
$$
\int_{\Omega} F(x,\sqrt{4m}v_n) \to 0
$$
as $n \to +\infty$ for any fixed $m>0$.

By the generality of constant $m$, without any loss of generality,
we suppose that $\sqrt{4m}<\|u_n\|$. Let $t_n \in [0,1]$ be in such way that
\begin{equation} \label{def_tn}
I(t_nu_n) = \max_{t \in [0,1]} I(tu_n).
\end{equation}
Using the definition of $t_n$ in \eqref{def_tn}, considering
$t = \sqrt{4m}<\|u_n\|$ we deduce that
\begin{equation}\label{m}
\begin{aligned}
I(t_nu_n)  &\geq  I \Big( \frac{\sqrt{4m}}{\|u_n\|} u_n\Big) \\
&= 2m - \int_{\Omega} F(x,\sqrt{4m}v_n)\, dx \
 \geq  m > 0,
\end{aligned}
\end{equation}
for any $n \geq n_0$, where $n_0 \in \mathbb{N}$ depends only on $m$.
It is important to emphasize that if $t_{n} = 0$ holds for any $n \in \mathbb{N}$
we obtain $0 = I(t_{n}u_{n}) \geq m > 0$ which is a contradiction.
 Furthermore, assuming that $t_{n} = 1$ holds for any $n \in \mathbb{N}$
we also obtain $m \leq I(t_{n}u_{n}) = I(u_{n})\to c$ which does not make
 sense for any $m > c$. Hence, up to a subsequence,
we can assume that $t_n \in (0,1)$. This assertion together with
\eqref{def_tn} imply that $I'(t_{n} u_{n})(t_{n} u_{n}) = 0$ for any
$n \in \mathbb{N}$.

Now we shall divide the proof in two cases:
\smallskip

\noindent\textbf{Case 1:} Along a subsequence we suppose that
$t_n < (2/\|u_n\|)$.
In this case we use hypotheses (H1)--(H3) and the Sobolev
embedding  to get $c_1,c_2>0$, satisfying the estimates
$$
\Big|\int_{\Omega} H(x,t_nu_n\,) dx \Big|
\leq c_1 (t_n\|u_n\|)^{2} + c_2 (t_n\|u_n\|)^p
\leq 4c_1 + c_2 2^p < \infty.
$$
Using the fact that $t_n \in (0,1)$ it follows from the identity
$I'(t_nu_n)(t_nu_n)=0$ that
\begin{equation*}\label{zero}
0 = t_n^2 \|u_n\|^2 - \int_{\Omega} f(x,t_nu_n)(t_nu_n)\, dx
= 2I(t_nu_n) - \int_{\Omega} H(x,t_nu_n)\, dx.
\end{equation*}
As a consequence we obtain
$$
I(t_nu_n) = \frac{1}{2} \int_{\Omega} H(x, t_{n} u_{n})\,dx \leq c_3
$$
where $c_{3} > 0$. This give us an absurd with \eqref{m} due the fact that $m>0$
in that expression is arbitrary. Hence Case 1 is not possible.
It remains to focus in the following case
\smallskip

\noindent\textbf{Case 2}: Along a subsequence we have
$t_n \geq (2/\|u_n\|)$.
In this case we consider a new sequence $s_{n} : = \frac{1}{\|u_{n}\|} < t_{n}$.
Analyzing the energy functional $I$ we obtain the  identity
\begin{align*}
\frac{I(t_{n}u_{n})}{t_{n}^{2} \|u_{n}\|^{2}}
- \frac{I(s_{n}u_{n})}{s_{n}^{2} \|u_{n}\|^{2} }
& =  \int_{\Omega} \frac{F(x,t_{n}u_{n})}{s_{n}^{2} \|u_{n}\|^{2}}
 \mathrm{d}x - \int_{\Omega} \frac{F(x,s_{n}u_{n})}{t_{n}^{2} \|u_{n}\|^{2}}
 \mathrm{d}x \\
& = -\frac{1}{\|u_{n}\|^{2}}\int_{\Omega}
 \Big(\frac{F(x,t_{n}u_{n})}{t_{n}^{2}}
 - \frac{F(x,s_{n}u_{n})}{s_{n}^{2}}\Big)\, \mathrm{d}x
\end{align*}
At this moment, using Calculus Fundamental Theorem, we infer that
\begin{align*}
\frac{I(t_{n}u_n)}{t_{n}^{2}\|u_n\|^2} - \frac{I(s_{n}u_n)}{s_{n}^2 \|u_n\|^2}
&=- \int_{\Omega} \int_{s_{n}}^{t_{n}} \frac{d}{d\tau} \Big(
\frac{F(x,\tau u_n)}{\tau^2\|u_n\|^2}\Big) \mathrm{d}\tau\,\mathrm{d}x\\
&= - \int_{\Omega} \int_{s_{n}}^{t_{n}} \frac{H(x,\tau u_n)}{\tau^3\|u_n\|^2}
\,\mathrm{d}\tau\,\mathrm{d}x.
\end{align*}


Now we shall define the function $\phi: \mathbb{R}\to \mathbb{R}$ given by
$\phi(t) = e^{-1/t^{2}}$, $t \neq 0$ and $\phi(0) = 0$. It follows
from elementary calculus that $\phi \in C^{\infty}(\mathbb{R}, \mathbb{R})$
and $0 \leq \phi(t) \leq 1$ for any $t \in \mathbb{R}$.
Furthermore, we observe that $\phi^{(j)}(0) = 0$ for any $j \in \mathbb{N}$.
Using the condition $(NQ)$ and the function $\phi$ given just above we infer that
\begin{equation}\label{H1}
 H(x,t) \geq R \phi(t), \quad \forall | t | \geq M,\;  x \in \Omega
\end{equation}
holds for any $R > 0$ where $M = M(R) > 0$.

On the other hand, by  hypothesis (H1) and continuity of the function $H$,
we have that
\begin{equation}\label{H2}
 H(x,t) \geq - C |t| , \quad \forall | t | \leq M,\; x \in \Omega
\end{equation}
for some positive constant $C = C(R) < \infty $.
 Using estimates \eqref{H1} and \eqref{H2} we ensure the global inequality
\begin{equation*}
 H(x,t) \geq R \phi(t) - C |t| , \quad \forall  (x,t) \in \Omega \times \mathbb{R}.
\end{equation*}
Therefore, we have
\begin{equation*}
\frac{I(t_{n}u_{n})}{t_{n}^{2} \|u_{n}\|^{2}}
- \frac{I(s_{n}u_{n})}{s_{n}^{2} \|u_{n}\|^{2}}\leq I_{1} + I_2,
\end{equation*}
where $I_{1} $ and $I_2$ are defined by
\begin{gather}\label{I1}
 I_{1}= -\frac{R}{\|u_{n}\|^{2}}\int_{\Omega}
\Big[ \int_{s_{n}}^{t_{n}} \frac{\phi(\tau u_{n})}{\tau^{3}} d\tau\Big]\, dx,\\
\label{I2}
I_2 = \frac{C}{\|u_{n}\|^{2}} \int_{\Omega}\Big[\int_{s_{n}}^{t_{n}}
\frac{| \tau u_{n} |}{\tau^{3}} d\tau\Big]\,dx.
\end{gather}
Firstly, analyzing the integral \eqref{I2} we easily see that
\begin{equation*}
 I_2 = C \int_{\Omega}\Big(
\frac{1}{s_{n}} - \frac{1}{t_{n}}\Big)\frac{|u_{n}|}{\|u_{n}\|^{2}}\,dx.
\end{equation*}
At this moment using the fact that $t_{n} > \frac{2}{\|u_{n}\|}$
the integral $I_2$ can be estimated as follows
\begin{equation*}
I_2 \leq \frac{C}{2} \int_{\Omega}|v_{n}|\,dx \to 0.
\end{equation*}
Now we shall analyze the integral in \eqref{I1}. According to mean value
theorem for integrals there exists $c_{n} \in (s_{n}, t_{n})$ in such way that
\begin{equation*}
\int_{s_{n}}^{t_{n}} \frac{\phi(\tau u_{n})}{\tau^{3}} \, d\tau
= \phi(c_{n} u_{n}) \int_{s_{n}}^{t_{n}} \frac{1}{\tau^{3}}\, d\tau
= - \phi(c_{n} u_{n}) ( 1/2 t_{n}^{2} - 1/2 s_{n}^{2} ) .
\end{equation*}
As a consequence we get the  identity
\begin{equation*}
 I_{1}= \frac{R}{\|u_{n}\|^{2}}\Big(\frac{1}{t_{n}^2} - \frac{1}{s_{n}^2}\Big)
\int_{\Omega} \phi(c_{n} u_{n})\,dx.
\end{equation*}
The above identity implies that
$$
{I_{1} \leq -\frac{3R}{8}\int_{\Omega} \phi(c_{n} u_{n}) \,dx}.
$$

Now we define the set
\begin{equation*}
\Omega_{n} =  \cup_{M > 0} \big\{x \in \Omega : |(c_{n}u_{n})(x)|
\geq \frac{1}{M}\big\}.
\end{equation*}
The strategy here is to find a subsequence $(u_{n_{k}}) \in \mathcal{H} $
in such way that
\begin{equation}\label{aff}
|\Omega_{n_{k}}| \geq \delta_{0} > 0, \quad \forall  k \in \mathbb{N}
\end{equation}
holds for some numbers $\delta_{0} > 0, M_{0} > 0$ fixed.
If this is true, then
\begin{equation*}
I_{1} \leq -\frac{3R}{8}\int_{\Omega_{n_{k}}} e^{-M_{0}^{2}}\, dx
\leq -\frac{3R}{8} \delta_{0} e^{-M_{0}^{2}}< 0.
\end{equation*}
Now we proceed with the proof of inequality \eqref{aff}.
Suppose by contradiction that for any subsequence $(u_{n_{k}}) \in \mathcal{H}$
and for each $\delta > 0$ and $M > 0$ we have
\begin{equation*}
0 < |\Omega_{n_{k}}| \leq \delta.
\end{equation*}
These inequalities show that $|\Omega_{n_{k}}| = 0$ for any $M > 0$.
Consider the measurable set $\Omega_{0} = \cup_{k = 1}^{\infty}\Omega_{n_{k}}$.
It is easy to verify that
\begin{equation}\label{Omega0}
|\Omega_{0}| = \big|\cup_{k = 1}^{\infty}\Omega_{n_{k}} \big|
\leq \sum_{k = 1}^{\infty} |\Omega_{n_{k}}|= 0.
\end{equation}
Note that we can rewrite the set
$\Omega = \Omega_{0}\, \dot{\cup} (\Omega\setminus \Omega_{0})$.
It is important to emphasize that
\begin{equation*}
I(t_{n_{k}}u_{n_{k}}) = \frac{1}{2} \int_{\Omega} H (x,t_{n_{k}}u_{n_{k}})\, dx
+ o_{k}(1)
\end{equation*}
Using the previous identity and \eqref{Omega0} we easily see that
\begin{equation}\label{Ik}
\begin{aligned}
I(t_{n_{k}}u_{n_{k}})
& =  \frac{1}{2}\Big(\int_{\Omega_{0}} H (x,t_{n_{k}}u_{n_{k}})\, dx
  \int_{\Omega\setminus \Omega_{0}} H (x,t_{n_{k}}u_{n_{k}})\, dx\Big) + o_{k}(1) \\
& =  \frac{1}{2} \int_{\Omega\setminus \Omega_{0}} H (x,t_{n_{k}}u_{n_{k}})\, dx
+ o_{k}(1)
\end{aligned}
\end{equation}
Analyzing the elements in the set $\Omega\setminus \Omega_{0}$ and using the
 Morgan's Law we get
\begin{equation*}
\Omega\setminus \Omega_{0} = \big\{ x \in \Omega : |(c_{n_{k}}u_{n_{k}})(x)|
< \frac{1}{M},\;  \forall  k \in \mathbb{N}, \; \forall  M > 0 \big\}.
\end{equation*}

Now we consider the  set
$\Omega_{1}= \{ x \in \Omega ; |(c_{n_{k}}u_{n_{k}})(x)| = 0 \;\forall k \in \mathbb{N}
\}$. Clearly, we see that $\Omega_{1} \subset \Omega\setminus \Omega_{0}$.
 Additionally, given any $x \in \Omega\setminus \Omega_{0}$, we observe that
$$
|(c_{n_{k}}u_{n_{k}})(x)| < \frac{1}{M}, \quad \forall  k \in \mathbb{N}
$$
holds  for any $M > 0$. As the inequality just above is verified for any $M > 0$
and for any integer $k$ we obtain that
$|(c_{n_{k}}u_{n_{k}})(x)| = 0$ for all $k \in \mathbb{N}$. As a consequence
 $x \in \Omega_{1}$ and $ \Omega\setminus \Omega_{0} = \Omega_{1}$.
Furthermore, using the fact that $0 < s_{n_{k}} \leq c_{n_{k}} \leq t_{n_{k}}$,
we mention also that
\begin{equation*}
u_{n_{k}}(x) = 0, x \in \Omega\setminus \Omega_{0},\; \forall  k \in \mathbb{N}.
\end{equation*}
Thus, we can be rewritten the set $\Omega\setminus \Omega_{0}$ in the  form
\begin{equation*}
\Omega\setminus \Omega_{0} = \{ x \in \Omega; |u_{n_{k}}(x)| = 0 \; \forall
 k \in \mathbb{N} \}.
\end{equation*}

From now on, using the estimate given in \eqref{Ik} and the above assertion,
 we assume that
\begin{equation*}
I(t_{n_{k}}u_{n_{k}})
= \frac{1}{2} \int_{\Omega\setminus \Omega_{0}} H (x,t_{n_{k}}u_{n_{k}})\, dx
+ o_{k}(1) = o_{k}(1),
\end{equation*}
which implies the  convergence
\begin{equation*}
\lim_{k \to +\infty} I(t_{n_{k}}u_{n_{k}}) = 0.
\end{equation*}
This is a contradiction with \eqref{m} for each $m > 0$.
Therefore the assertion given in \eqref{aff} is  true.
In conclusion, we have been proved that $v \neq 0$, i.e, the set
$$
\widehat{\Omega} = \{ x \in \Omega : v(x) \neq 0\}
$$
has positive Lebesgue measure. Additionally, we know that $|u_{n}| \to \infty$
a.e. in $\widehat{\Omega}$. Taking into account hypothesis (NQ) we observe that
$H(x, t) \geq - C$ holds for any $(x, t) \in \Omega \times \mathbb{R}$ and
for some $C > 0$.
Under these conditions it follows from Fatou's Lemma that
\begin{align*}
c &= \liminf_{n \to \infty} \big\{ I(u_{n}) - \frac{1}{2} I'(u_{n})u_{n}\big\} \\
&= \frac{1}{2} \liminf_{n \to \infty} \int_{\Omega} H(x, u_{n}) dx  \\
&\geq \frac{1}{2} \int_{\Omega} \liminf_{n \to \infty} H(x, u_{n}) dx = \infty.
\end{align*}
Hence we have a contradiction proving that $(u_{n}) \in \mathcal{H}$
is now a bounded sequence. This finishes the proof of Proposition \ref{cerami}.
\end{proof}


\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1}]
Initially, using hypotheses (H1)--(H3) and the fact that
$ f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R})$, we have estimate
\begin{equation}\label{crescimento de F}
| F(x,t) | \leq \frac{(\varepsilon + f_{0})}{2}|t| ^{2} + C |t| ^{p},\quad
\forall (x,t) \in {\Omega}\times \mathbb{R},
\end{equation}
for any $\varepsilon > 0$ and for some $C = C (\varepsilon) > 0$.
Now we mention the following variational inequality
\begin{equation}\label{v1}
\|u\|_2^{2} \leq \frac{1}{\mu_{1}} \|u\|^{2}, \quad u \in \mathcal{H}.
\end{equation}
Noticing the definition of $I$ given in \eqref{Funcional}, it follows
from \eqref{crescimento de F}, \eqref{v1} and Sobolev inequality that
\begin{equation*}
I(u) \geq \Big(\frac{1}{2} - \frac{(\varepsilon + f_{0})}{2\mu_{1}}\Big) \| u\| ^{2}
 - C \| u\| ^{p}.
\end{equation*}
for some positive constant $C > 0$. Now we define
\[
r = \left( \frac{\varepsilon_{0}}{4 \mu_{1}C}\right)^{1/(p-2)},\quad
\rho = r^{2} \left( \frac{\varepsilon_{0}}{4\mu_{1}}\right)
\]
where $\varepsilon_{0} = \frac{\mu_{1} - f_{0}}{2} > 0$. Under these conditions,
for any $\varepsilon \in (0, \varepsilon_{0})$, we infer that
$$
I(u) \geq \rho > 0,\quad \forall u \in \mathcal{H},\; \| u\| =  r.
$$
This shows the first statement in the mountain pass geometry given
in Theorem \ref{mountain}.

Now using condition (H2) and the continuity of $F$, given any $R > \mu_{1}$
there exists $C_{R} < \infty$ in such way that
\begin{equation}\label{pm2}
F(x,t) \geq \frac{R t^{2}}{2} - C_{R}, \quad \forall  (x,t) \in \Omega
 \times \mathbb{R}.
\end{equation}
Let $\varphi_{1} > 0$ be the eigenfunction associated to $\mu_{1}$ which
satisfies $\int_{\Omega} \varphi_{1}^{2} dx = 1$. Consider
$e = t \varphi_{1}$ with $t>0$. Using \eqref{pm2} we get the  estimate
\begin{equation*}
I(e) = I(t\varphi_{1}) \leq \frac{t^{2}}{2}(\mu_{1} - R) + C_{R}|\Omega|.
\end{equation*}
As a consequence there exists $t_{0} > 0$ large enough in such way that,
considering $ R > \mu_{1}$, we obtain $\|e\| = \|t_{0}\varphi_{1}\| > r$
and $I(e) < 0$. These facts shows that $I$ admits the mountain pass geometry
proving the existence of a Cerami condition $(u_{n})$ at the mountain
pass level given by \eqref{nivel}. According to Proposition \ref{cerami}
there exists $u \in \mathcal{H}$ in such way that $u_{n} \to u$ in
$\mathcal{H}$. As a consequence $u$ is a critical point for $I$
and $I(u) \geq \rho > 0$. Hence $u$ is a weak solution to the elliptic
problem \eqref{pi}. Here was used the fact that $u = 0$ and $\Delta u = 0$
on $\partial \Omega$ which we shall prove in the Appendix.
This completes the proof.
\end{proof}


\begin{proof}[Proof of theorem \ref{thm2}]
In this section, via Strong Maximum Principle, we shall prove that
problem \eqref{pi} admits at least two solutions which one solution
is positive and another one is negative. To do that we reduce the fourth-order
problem \eqref{pi} to a second-order elliptic problem.
Hence, using a truncation argument, we analyze the positive part and negative
part of the nonlinearity $f$ obtaining a multiplicity result.

As a first step we shall consider $v = - \Delta u$ in problem \eqref{pi},
i.e, we reduce problem \eqref{pi} into the  elliptic system
\begin{equation}\label{p2}
\begin{gathered}
- \Delta u  =  v \quad  \text{in } \Omega, \\
 - \alpha \Delta v - \beta v  =  f(x,u)\quad  \text{in } \Omega, \\
 u = v  =  0 \quad \text{on }  \partial \Omega.
\end{gathered}
\end{equation}
Here we emphasize that $u, v \in H_0^1(\Omega)$. Putting $v^{-} \in H_0^1(\Omega) $ as test
function in the problem \eqref{p2} we observe that
\begin{equation}\label{v-}
\int_{\Omega} \alpha \nabla v \nabla v^{-} \, dx
- \beta \int_{\Omega} v v^{-} \, dx = \int_{\Omega} f(x,u)\, v^{-} \, dx.
\end{equation}
It is worthwhile to mention that
\begin{equation*}
\nabla v^{-} = \begin{cases}
\nabla v, &  \text{if } v < 0 \\
0, &  \text{if } v \geq 0,
\end{cases}
\end{equation*}
Using the variational inequality for $\lambda_{1}$ and the identity \eqref{v-}
 we obtain
\begin{equation*}
\Big( \alpha - \frac{\beta}{\lambda_{1}}\Big)\int_{\Omega} |\nabla v^{-}|^{2} \, dx
\leq \int_{\Omega} \alpha |\nabla v^{-}|^{2} \, dx - \beta \int_{\Omega} |v^{-}|^{2} \, dx
 = \int_{\Omega} f(x,u)\, v^{-} \, dx .
\end{equation*}
At this moment we shall consider the function $f^{+}$, i.e, we define the
truncation
\begin{equation*}
 f^{+}(x,t) = \begin{cases}
f(x,t), &  \text{if } t \geq 0 \\
0, & \text{if } t < 0.
\end{cases}
\end{equation*}
Now we define the functional $I^{+}: \mathcal{H} \to \mathbb{R}$ of $C^{1}$
class given by
\begin{equation*}
I^{+}(u) = \frac{\alpha}{2} \int_{\Omega} |\Delta u|^{2} dx
- \frac{\beta}{2} \int_{\Omega} |\nabla u|^{2} dx
- \int_{\Omega} F^{+}(x, u) \, dx
\end{equation*}
where $F^{+}(x,t) = \int_{0}^{t} f^{+}(x,s) ds$ for any
$x \in \Omega$, $t \in \mathbb{R}$. Moreover, $u \in \mathcal{H}$ is a critical
point for $I^{+}$ if and only if we have
\begin{equation*}
\alpha \int_{\Omega} \Delta u \Delta \phi dx
 - \beta \int_{\Omega} \nabla u \nabla \phi dx
- \int_{\Omega} f^{+}(x, u) \phi \, dx = 0
\end{equation*}
for any $\phi \in \mathcal{H}$. Then we can find weak solution to the elliptic
 problem \eqref{pi} finding positive critical points to the functional $I^{+}$.
 It is no hard to verify that $I^{+}$ admits the mountain pass geometry
using the same ideas discussed in the proof of Theorem \ref{thm1}.
Furthermore, the functional $I^{+}$ satisfies the $(Ce)_{c}$ condition at any
energy level $c \in \mathbb{R}$. This is ensured using the ideas discussed
in the proof of Proposition \ref{cerami}. As a consequence, using the
 Mountain Pass Theorem, there exists a critical point $u \in \mathcal{H}$
for the functional $I^{+}$ verifying $I^{+}(u) > 0$.

Now we shall consider the  auxiliary elliptic problem
\begin{equation}\label{p22}
\begin{gathered}
- \Delta u  = v \quad \text{in } \Omega \\
 - \alpha \Delta v - \beta v  =  f^{+}(x,u)\quad \text{in } \Omega \\
 u = v  =  0 \quad \text{on } \partial \Omega.
\end{gathered}
\end{equation}
Using the same ideas discussed above and changing the function $f$ by
$f^{+}$ we get the  estimate
\begin{equation}\label{v-2}
0 \leq \Big( \alpha - \frac{\beta}{\lambda_{1}}\Big)\int_{\Omega} |\nabla v^{-}|^{2} \, dx
\leq \int_{[v < 0]} f^{+}(x,u)\, v^{-} \, dx.
\end{equation}
It follows from problem \eqref{p22} that
\begin{equation}\label{e1}
\begin{gathered}
- \Delta u  =  v < 0 \quad \text{in } [v < 0 ] \\
 u \leq  0 \quad \text{on }  \partial [v < 0 ],
\end{gathered}
\end{equation}
where we define $[v < 0 ] = \{ x \in \Omega : v(x) < 0 \}$.
This set is an open set due the fact that $v$ is in
 $C^{0,\alpha}(\overline{\Omega})$. This fact can be ensured using
regularity arguments on elliptic equations involving operators of fourth-order
and the fact that $f^{+}$ is a continuous function. For further results
on regularity for elliptic equations we infer the reader to
 Agmon, Douglis and Niremberg \cite{ADN} or Gupta and Kwong \cite{Gupta}.

Now, using the strong maximum principle \cite{Gilbard}, we note
that \eqref{e1} implies $u < 0$ in $[v < 0 ]$.
As a consequence, using the last assertion, we know that
\begin{align*}
\int_{\Omega} f^{+}(x,u)\, v^{-} \, dx
&= \int_{[v \geq 0]} f^{+}(x,u)\, v^{-} \, dx
 + \int_{[v < 0]} f^{+}(x,u)\, v^{-} \, dx  \\
&= \int_{[v < 0]} f^{+}(x,u)\, v^{-} \, dx = 0
\end{align*}
Therefore,  estimate \eqref{v-2} and the variational inequality for
$\lambda_{1}$ imply
\begin{equation*}
0 \leq \Big( \alpha - \frac{\beta}{\lambda_{1}}\Big)
\int_{\Omega} |\nabla v^{-}|^{2} \, dx \leq 0.
\end{equation*}
Hence $v^{-} \equiv 0$ and $v = v^{+} \geq 0$ in $\Omega$.
 As a consequence,
\begin{equation*}
- \Delta u = v \geq 0  \text{ in } \Omega,\quad  u = 0 \text{ on }  \partial \Omega.
\end{equation*}
Using one more time the strong maximum principle we easily seen that
 $u > 0 $ in $\Omega$, i.e, we guarantee that problem \eqref{pi} admits
at least one positive solution.

Analogously, we define the function $f^{-}$  by
\begin{equation*}
 f^{-}(x,t) = \begin{cases}
f(x,t), &  \text{if }  t \leq 0 \\
0, &  \text{if } t > 0.
\end{cases}
\end{equation*}
Now we define the functional $I^{-}: \mathcal{H} \to \mathbb{R}$ of $C^{1}$
class given by
\begin{equation*}
I^{-}(u) = \frac{\alpha}{2} \int_{\Omega} |\Delta u|^{2} dx
- \frac{\beta}{2} \int_{\Omega} |\nabla u|^{2} dx
- \int_{\Omega} F^{-}(x, u) \, dx
\end{equation*}
where $F^{-}(x,t) = \int_{0}^{t} f^{-}(x,s) ds$ for any
$x \in \Omega, t \in \mathbb{R}$. Moreover, $u \in \mathcal{H}$
is a critical point for $I^{-}$ if only if we have
\begin{equation*}
\alpha \int_{\Omega} \Delta u \Delta \phi dx
- \beta \int_{\Omega} \nabla u \nabla \phi dx
- \int_{\Omega} f^{-}(x, u) \phi \, dx = 0
\end{equation*}
for any $\phi \in \mathcal{H}$. Then we can find weak solution to the
 elliptic problem \eqref{pi} finding negative critical points to
the functional $I^{-}$. It is no hard to verify that $I^{-}$ admits
the mountain pass geometry using the same ideas discussed in the proof
of Theorem \ref{thm1}. Furthermore, the functional $I^{-}$ satisfies the
$(Ce)_{c}$ condition at any energy level $c \in \mathbb{R}$.
This is verified using the ideas discussed in the proof of
Proposition \ref{cerami}. As a consequence, using the Mountain Pass Theorem,
there exists a critical point $w \in \mathcal{H}$ for the functional $I^{-}$
satisfying $I^{-}(w) > 0$. At the same time, using the strong principle
strong maximum twice we obtain a second solution $w \in \mathcal{H}$ to the
elliptic problem \eqref{pi} satisfying $w < 0$ in $\Omega$. Here was used
the fact that $f^{-}$ is also a continuous function. This completes the proof.
\end{proof}

To prove Theorem \ref{thm3},
we shall apply the following version of the symmetric Mountain Pass Theorem.

\begin{theorem}\label{theomSMP}
Let $X$ be an infinite dimensional Banach space and 
$\mathcal{I} \in C^{1}(X,\mathbb{R})$ be even, satisfy $(Ce)_{c}$ for any
$c \in \mathbb{R}$, and $\mathcal{I}(0) = 0$. If $X = V \oplus W$, where $V$
is finite dimensional, and $\mathcal{I}$ satisfies
\begin{itemize}
\item[(1)] there exists $r, \rho > 0$ such that
$$ 
\mathcal{I}(u) \geq r > 0,\quad \text{for any } u \in \partial B_{\rho}(0)\cap W;
$$

\item[(2)] for any finite dimensional subspace $\overline{X} \subset X$ there exists 
$\xi = \xi(\overline{X})$ such that
$$ 
\mathcal{I}(u) \leq 0 \quad \text{for any } u 
\in \overline{X}\setminus B_{\xi}(0),
$$
\end{itemize}
then $\mathcal{I}$ possesses an unbounded sequence of critical values.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thm3}]
 Firstly, we would like to apply Theorem \ref{theomSMP} choosing 
$X = \mathcal{H}$ and $\mathcal{I} = I$. Hence, using hypotheses 
$(f_0) - (f_2)$, we deduce the  estimate
\begin{equation*}
| F(x,t) | \leq \frac{(\varepsilon + f_{0})}{2}|t| ^{2} 
+ C |t| ^{p},\quad\forall  (x,t) \in {\Omega} \times \mathbb{R},
\end{equation*}
which holds for any $\varepsilon > 0$ and for some $C = C (\varepsilon) > 0$. 
This implies that $I(u) \geq \rho $ for any $u \in \mathcal{H}$ satisfying 
$\|u\| = r$. Here we choose $r, \rho > 0$ small, see the proof of 
Theorem \ref{thm1}, and therefore (1) of Theorem \ref{theomSMP} holds.

Now consider any subspace $E \subset \mathcal{H}$ which is finite dimensional. 
Let $B_{r}(0) \subset \mathcal{H}$ be the open ball centered at the origin 
and radius $r > 0$. Here we is need to guarantee that $I(v) \leq 0$ for any 
$v \in E \backslash {B_{r}(0)}$ where $r > 0$ is chosen large enough. 
To do that, we use the estimate \eqref{pm2} and the fact that all the norms in 
space $E$ are equivalent obtaining constants $C_{0} > 0$ and $C_{R} > 0$ 
in such way that
\begin{equation*}
I(v) \leq \frac{1}{2}\left(1- C_{0} R \right)\|u\|^{2} 
+ C_{R}|\Omega|,\quad  v \in E
\end{equation*}
holds for any $R > 0$. Since $R > 0$ is arbitrary we conclude that 
$I(v) \to - \infty $ as $\|v\|\to \infty$, $v \in E $, and therefore 
 (1) of Theorem \ref{theomSMP} holds.
Furthermore, the functional $I$ satisfies the Cerami condition, 
see Proposition \ref{cerami}. Then applying the symmetric mountain pass 
theorem we obtain a sequence of critical values $c_{n} > 0$ 
such that $c_{n} \to + \infty$ as $n \to \infty$. 
In particular, we obtain a sequence of critical points $u_{n} \in \mathcal{H}$ 
for the functional $I$ satisfying $I(u_{n}) \to + \infty$ as $n \to \infty$. 
This completes the proof.
\end{proof}

\section{Appendix}

The main objective in this section is to ensure that any critical point 
$u \in \mathcal{H}$ for $I$ satisfies $u = 0$ and $\Delta u = 0$ on 
$\partial \Omega$. In other words, we shall prove that any critical point 
of $I$ give us a weak solution for the problem \eqref{pi} satisfying the
 Navier boundary conditions.

Let $u \in \mathcal{H} = H_0^1(\Omega)\cap H^2(\Omega)$ be a critical point for $I$, i.e, we have that
\begin{equation*}
 \alpha \int_{\Omega} \Delta u \Delta \phi\, dx + \beta \int_{\Omega} \phi \Delta u \, dx 
= \int_{\Omega} f(x,u) \phi \, dx.
\end{equation*}
holds for any function $\phi \in \mathcal{H}$. From standard trace theorem 
we know that $u = 0$ on $\partial \Omega$.
It remains to show that $\Delta u = 0$ on $\partial \Omega$. 
To do that we consider $ v = - \Delta u $ and $h(x) = f(x,u)$. 
It is easy to verify that $v \in L^{2}(\Omega)$ and $h \in L^{2}(\Omega)$.
 Furthermore, we have
\begin{equation}\label{01}
 \int_{\Omega} v[-\alpha \Delta \phi -\beta \phi]\, dx 
= \int_{\Omega} h(x) \phi \, dx,\quad  \forall \phi \in \mathcal{H}.
\end{equation}
Let $w \in H_0^1(\Omega)\cap H^2(\Omega)$ be the unique weak solution for the elliptic problem
\begin{equation}\label{prow}
\begin{gathered}
- \alpha \Delta w -\beta w  =  h(x) \quad \text{in }\Omega \\
 w  =  0 \quad \text{on }  \partial \Omega.
\end{gathered}
\end{equation}
To ensure the existence of solutions to the elliptic problem \eqref{prow} 
for each $h \in L^{2}(\Omega)$ we apply standard minimization procedures.
To get the uniqueness of solutions to the elliptic problem \eqref{prow} 
we consider any weak solutions $w_{1}$ and $w_2$ proving that
\begin{gather*}
- \alpha \Delta(w_{1} - w_2) -\beta (w_{1} - w_2)  =  0 \quad \text{in }\Omega \\
 (w_{1} - w_2)  =  0 \quad \text{on }  \partial \Omega.
\end{gather*}
Choosing $\overline{w} = w_{1} - w_2$ and taking $\overline{w}$ as testing 
function we get
\begin{equation*}
\int_{\Omega} \alpha |\nabla \overline{w} |^{2} - \beta \overline{w}^{2} \, dx = 0.
\end{equation*}
On the other hand, using the hypothesis $ \mu_{1}> 0$, we deduce that 
$-\infty < \beta < \alpha\lambda_{1}$. In particular, putting $\beta \leq 0$, 
it follows from the above estimate that
\begin{equation*}
\int_{\Omega} \alpha |\nabla \overline{w} |^{2}\, dx \leq 0.
\end{equation*}
This implies that $\overline{w} \equiv 0$. Now, taking 
$0 < \beta < \alpha \lambda_{1}$, the variational inequality says
\begin{equation*}
0 \leq \int_{\Omega} \Big(\alpha - \frac{\beta}{\lambda_{1}}\Big) |\nabla \overline{w} |^{2}\, dx
 \leq 0.
\end{equation*}
Hence the last estimate implies that $\overline{w} \equiv 0$. To sum up, we 
have been shown that $w_{1} \equiv w_2$ in $\Omega$ proving that
problem \eqref{prow} admits exactly one weak solution for each $h \in L^{2}(\Omega)$.

At this moment, using the weak formulation for \eqref{prow} we obtain
\begin{equation}\label{02}
\int_{\Omega} - \alpha w \Delta\phi - \beta w \phi \, dx = \int_{\Omega} h(x)\phi\, dx
\end{equation}
holds for any $\phi \in \mathcal{H}$. Putting together the identities \eqref{01} 
and \eqref{02} we deduce that
\begin{equation*}
\int_{\Omega} (v - w)(\alpha \Delta\phi - \beta \phi )\, dx = 0, \quad
 \forall \phi \in C_{0}^{\infty}(\Omega).
\end{equation*}
At this stage, using Du Bois Raymond's Lemma \cite{Gilbard}, we infer that 
$v = w$ a.e. in $\Omega$.
As a consequence $v = w $ in $L^{2}(\Omega)$ which says that 
$v = - \Delta u = 0$ on $\partial \Omega$.


\subsection*{Acknowledgments}
This research was supported by Fapeg/CNpq grant 03/2015-PPP.
The authors would like to thank the anonymous referee for the carefully reading 
of the earlier version for this work, and for giving us many constructive and 
interesting remarks on fourth-order elliptic problems.

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\end{document}