\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 117, pp. 1--10.\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-2017/117\hfil Existence of infinitely many solutions] {Existence of infinitely many solutions for fourth-order equations depending on two parameters} \author[A. Hadjian, M. Ramezani \hfil EJDE-2017/117\hfilneg] {Armin Hadjian, Maryam Ramezani} \address{Armin Hadjian (corresponding author)\newline Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran} \email{hadjian83@gmail.com, a.hadjian@ub.ac.ir} \address{Maryam Ramezani \newline Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran} \email{mar.ram.math@gmail.com} \dedicatory{Communicated by Vicentiu Radulescu} \thanks{Submitted December 21, 2016. Published May 3, 2017.} \subjclass[2010]{34B15, 58E05} \keywords{Fourth-order equations; variational methods; \hfill\break\indent infinitely many solutions} \begin{abstract} By using variational methods and critical point theory, we establish the existence of infinitely many classical solutions for a fourth-order differential equation. This equation has nonlinear boundary conditions and depends on two real parameters. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} The aim of this article is to study the fourth-order problem \begin{equation}\label{1.1} \begin{gathered} u^{(iv)}(x)= \lambda f(x,u(x)) \quad \text{in } [0,1],\\ u(0)=u'(0)=0,\\ u''(1)=0,\quad u'''(1)=\mu g(u(1)), \end{gathered} \end{equation} where $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is an $L^1$-Carat\'{e}odory function, $g:\mathbb{R}\to \mathbb{R}$ is a continuous function and $\lambda, \mu$ are positive parameters. Problem \eqref{1.1} describes the static equilibrium of a flexible elastic beam of length 1 when, along its length, a load $f$ is added to cause deformation. Precisely, conditions $u(0)=u'(0)=0$ mean that the left end of the beam is fixed and conditions $u''(1)=0$, $u'''(1)=\mu g(u(1))$ mean that the right end of the beam is attached to a bearing device, given by the function $g$. Existence and multiplicity of solutions for fourth-order boundary value problems has been discussed by several authors in the last decades; see for example \cite{BR,BCT,BD,CT,DM,LZL,M,S,TC,YCY} and references therein. Yang et al.\ \cite{YCY}, used Ricceri's variational principle \cite{Ricceri2} to establish the existence of at least two classical solutions generated from $g$ for problem \eqref{1.1}, with $\mu=1$. The authors in \cite{CT}, using a multiplicity result by Cabada and Iannizzotto \cite{CI}, ensured the existence of at least two nontrivial classical solutions for the problem \begin{gather*} u^{(4)}(x)+\lambda f(x,u(x))=0, \quad 0\frac{2^7\pi^4}{27B^\star}$, for every nonpositive continuous function $g:\mathbb{R}\to\mathbb{R}$ satisfying the condition \[ g_\infty:=\limsup_{\xi\to+\infty}\frac{-\int_0^\xi g(t)dt}{\xi^2} <+\infty, \] and for each $\mu\in\big]0,\frac{1}{2g_\infty}\big[$, the problem \begin{equation}\label{e3.6} \begin{gathered} u^{(iv)}(x)= \lambda f(u(x)) \quad \text{in } [0,1],\\ u(0)=u'(0)=0,\\ u''(1)=0,\quad u'''(1)=\mu g(u(1)), \end{gathered} \end{equation} admits infinitely many classical solutions. \end{theorem} The plan of the article is as follows. In Section 2 we introduce our notation and a suitable abstract setting (see Lemma \ref{lem 2.1}). In Section 3 we present our main result (see Theorems \ref{t3.1} and \ref{t3.6}) and some significative consequences (see Theorem \ref{t3.5} as well as Corollaries \ref{cor3.3}, \ref{cor3.4} and \ref{c3.6}). A concrete example of an application is exhibited in Example \ref{ex3.3}. In the conclusion, we cite a recent monograph by Krist\'aly, R\u adulescu and Varga \cite{KrisRadVar} as a general reference on variational methods adopted here. \section{Preliminaries} We shall prove our results applying the following smooth version of \cite[Theorem 2.1]{BonaMolica}, which is a more precise version of Ricceri's variational principle \cite[Theorem 2.5]{Ricceri1}. We point out that Ricceri's variational principle generalizes the celebrated three critical point theorem of Pucci and Serrin \cite{PS1, PS2} and is an useful result that gives alternatives for the multiplicity of critical points of certain functions depending on a parameter. \begin{lemma}\label{lem 2.1} Let $X$ be a reflexive real Banach space, let $\Phi,\Psi:X\to \mathbb{R}$ be two G\^{a}teaux differentiable functionals such that $\Phi$ is sequentially weakly lower semicontinuous, strongly continuous and coercive, and $\Psi$ is sequentially weakly upper semicontinuous. For every $r>\inf_X\Phi,$ let \begin{gather*} \varphi(r):=\inf_{u\in\Phi^{-1}(-\infty,r)} \frac{\big(\sup_{v\in\Phi^{-1}(-\infty,r)}\Psi(v)\big)-\Psi(u)}{r-\Phi(u)}, \\ \gamma:=\liminf_{r\to +\infty}\varphi(r),\quad \delta:=\liminf_{r\to(\inf_X\Phi)^+}\varphi(r). \end{gather*} Then the following properties hold: \begin{itemize} \item[(a)] If $\gamma<+\infty,$ then for each $\lambda\in ]0,1/\gamma[$, the following alternative holds: either \begin{itemize} \item[(a1)] $I_\lambda:=\Phi-\lambda\Psi$ possesses a global minimum, or \item[(a2)] there is a sequence $\{u_n\}$ of critical points (local minima) of $I_\lambda$ such that $$ \lim_{n\to+\infty}\Phi(u_n)=+\infty. $$ \end{itemize} \item[(b)] If $\delta<+\infty$, then for each $\lambda\in ]0,1/\delta[$, the following alternative holds: either \begin{itemize} \item[(b1)] there is a global minimum of $\Phi$ which is a local minimum of $I_\lambda$, or \item[(b2)] there is a sequence $\{u_n\}$ of pairwise distinct critical points (local minima) of $I_\lambda$ that converges weakly to a global minimum of $\Phi$. \end{itemize} \end{itemize} \end{lemma} We also refer the interested reader to \cite{BMR,MR14,MR014,MR} and the references therein, in which Ricceri's variational principle and its variants have been successfully used to obtain the existence of multiple solutions for different boundary value problems. Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be an $L^1$-Carat\'{e}odory function and $g:\mathbb{R}\to \mathbb{R}$ be a continuous function. We recall that $f:[0,1]\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carath\'eodory function if \begin{itemize} \item[(a)] the mapping $x\mapsto f(x,\xi)$ is measurable for every $\xi\in\mathbb{R}$; \item[(b)] the mapping $\xi\mapsto f(x,\xi)$ is continuous for almost every $x\in [0,1]$; \item[(c)] for every $\rho>0$ there exists a function $l_\rho\in L^1([0,1])$ such that $$ \sup_{|\xi|\leq \rho}|f(x,\xi)|\leq l_{\rho}(x) $$ for almost every $x\in [0,1]$. \end{itemize} Corresponding to $f, g$ we introduce the functions $F, G$ as follows \[ F(x,\xi):=\int_0^\xi f(x,t)dt,\quad G(\xi):=-\int_0^\xi g(t)dt, \] for all $x\in [0, 1]$ and $\xi\in\mathbb{R}$. We consider the space \[ X:=\{ u\in H^2([0,1]): u(0)=u'(0)=0\}, \] where $H^2([0,1])$ is the Sobolev space of all function $u:[0,1]\to \mathbb{R}$ such that $u$ and its distributional derivative $u'$ are absolutely continuous and $u''$ belongs to $L^2([0,1])$. $X$ is a Hilbert space with the inner product \[ \langle u , v\rangle := \int_0^1 u''(t)v''(t)dt \] and the corresponding norm \[ \| u\|:= \Big( \int_0^1 (u''(t))^2 dt\Big)^{1/2}. \] It is easy to see that the norm $\|\cdot\|$ on $X$ is equivalent to the usual norm $$ \int_0^1\left(|u(t)|^2+|u'(t)|^2+|u''(t)|^2\right)dt. $$ It is well known that the embedding $X\hookrightarrow C^1([0,1])$ is compact and \begin{equation}\label{2.1} \| u\|_{C^1([0,1])} := \max \{ \|u\|_\infty , \| u'\|_\infty\}\leq \|u\| \end{equation} for all $u\in X$ (see \cite{YCY}). We say that $u\in X$ is a {\it weak solution} of problem \eqref{1.1} if \[ \int_0^1 u''(x)v''(x)dt -\lambda \int_0^1 f(x,u(x))v(x)dx+\mu g(u(1))v(1)=0 \] for all $v\in X$. By a {\it classical solution} of problem \eqref{1.1} we mean a function $u\in C^1([0,1])$ such that $u^{(iv)}(x)\in C([0,1])$ and the boundary conditions and the equation are satisfied in $[0,1]$. In \cite[Lemma 2.1]{YCY} it has been shown that the weak solutions are classical solutions of problem \eqref{1.1}. \section{Main results} Before introducing the main result, we define some notation. We put \[ A:=\liminf_{\xi\to+\infty}\frac{\int_0^1 \max_{|t|\leq\xi}F(x,t)dx}{\xi^2},\quad B:=\limsup_{\xi\to+\infty}\frac{\int_{3/4}^1 F(x,\xi)dx}{\xi^2}. \] \begin{theorem}\label{t3.1} Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be an $L^1$-Carat\'{e}odory function. Assume that \begin{itemize} \item[(A1)] $F(x,\xi)\geq 0$ for all $(x,\xi)\in [0,3/4]\times \mathbb{R}$; \item[(A2)] $A<\frac{27}{64\pi^4} B$. \end{itemize} Then, for every $\lambda\in\Lambda:=]\frac{32\pi^4}{27B},\frac 1{2A}[$ and for every continuous function $g:\mathbb{R}\to \mathbb{R}$, whose potential $G$ satisfying the conditions $\inf_{\xi>0}G(\xi)=0$ and \begin{equation}\label{3'.1} G_\infty:=\limsup_{\xi\to+\infty}\frac{\max_{|t|\leq\xi}G(t)}{\xi^2}<+\infty, \end{equation} if we put \[ \mu_{G,\lambda}:=\frac{1}{2G_\infty}(1-2A\lambda), \] where $\mu_{G,\lambda}=+\infty$ when $G_\infty=0$, problem \eqref{1.1} has an unbounded sequence of classical solutions for every $\mu\in]0,\mu_{G,\lambda}[$ in $X$. \end{theorem} \begin{proof} Our aim is to apply Lemma \ref{lem 2.1}(a) to problem \eqref{1.1}. To this end, fix $\bar{\lambda}\in \Lambda$ and $g$ satisfying our assumptions. Since $\bar{\lambda}<\frac 1{2A}$, we have \[ \mu_{G,\bar{\lambda}}:=\frac{1}{2G_\infty}\left(1-2A\bar{\lambda}\right)>0. \] Now fix $\bar{\mu}\in]0,\mu_{G,\bar{\lambda}}[$. For each $u\in X$, let the functionals $\Phi,\Psi_{\bar{\lambda},\bar{\mu}}:X\to \mathbb{R}$ be defined by \begin{gather*} \Phi(u):=\frac{1}{2}\|u\|^2, \\ \Psi_{\bar{\lambda},\bar{\mu}}(u):=\int_0^1 F(x,u(x))dx+\frac{\bar{\mu}}{\bar{\lambda}}G(u(1)) \end{gather*} and put \[ I_{\bar{\lambda},\bar{\mu}}(u):=\Phi(u)-\bar{\lambda}\Psi(u),\quad u\in X. \] By standard arguments, it follows that $\Phi$ is sequentially weakly lower semicontinuous, strongly continuous and coercive. Moreover, $\Phi,\Psi_{\bar{\lambda},\bar{\mu}}\in C^1(X,\mathbb{R})$ and for any $u,v\in X$, we have \begin{gather*} \Phi'(u)(v)=\int_0^1u''(x)v''(x)dx, \\ \Psi_{\bar{\lambda},\bar{\mu}}'(u)(v)= \int_0^1 f(x,u(x))v(x)dx-\frac{\bar{\mu}}{\bar{\lambda}} g(u(1))v(1). \end{gather*} In \cite{YCY} the authors proved that $\Psi_{\bar{\lambda},\bar{\mu}}'$ is compact. Hence $\Psi_{\bar{\lambda},\bar{\mu}}$ is sequentially weakly (upper) continuous (see \cite[Corollary 41.9]{Zeidler}). First of all, we show that $\bar{\lambda}<1/\gamma$. Hence, let $\{\xi_n\}$ be a sequence of positive numbers such that $\lim_{n\to +\infty} \xi_n=+\infty$ and \[ \lim_{n\to +\infty} \frac{\int_0^1 \max_{|t|\leq \xi_n} F(x,t) dx}{\xi_n^2}=A. \] Put $r_n:={\xi_n^2}/{2}$ for all $n\in\mathbb{N}$. Then, for all $v\in X$ with $\Phi(v)0$ such that $\lim_{n\to +\infty} d_n=+\infty$ and \begin{equation}\label{3.1} \frac{1}{\bar{\lambda}}< \tau<\frac{27}{32\pi^4 } \frac{\int_{3/4}^1 F(x,d_n)dx}{d_n^2} \end{equation} for all $n\in\mathbb{N}$ large enough. For $n\in\mathbb{N}$ we define \begin{equation}\label{test} w_n(x):=\begin{cases} 0,& x\in [0,3/8],\\ d_n \cos^2 (4\pi x/3),& x\in ]3/8 , 3/4[,\\ d_n, & x\in [3/4, 1]. \end{cases} \end{equation} For any fixed $n\in\mathbb{N}$, it is easy to see that $w_n\in X$ and, in particular, one has \begin{equation}\label{3.2} \Phi(w_n)= \frac{32}{27} \pi^4 d_n^2. \end{equation} On the other hand, bearing (A1) and $\inf_{\xi>0}G(\xi)=0$ in mind, from the definition of $\Psi_{\bar{\lambda},\bar{\mu}}$ and \eqref{3.1}, we infer that \begin{equation}\label{3.3} \Psi_{\bar{\lambda}, \bar{\mu}}(w_n)\geq \int_{3/4}^1F(x,d_n) dx+\frac{\bar{\mu}}{\bar{\lambda}} G(d_n) \geq \frac{32}{27} \pi^4 \tau d_n^2. \end{equation} It follows from \eqref{3.2} and \eqref{3.3} that \[ I_{\bar{\lambda},\bar{\mu}} (w_n) \leq \frac{32}{27} \pi^4 d_n^2-\frac{32}{27} \pi^4\bar{\lambda}\tau d_n^2 = \frac{32}{27} \pi^4 (1-\bar{\lambda} \tau) d_n^2 \] for all $n\in\mathbb{N}$ large enough. Since $\bar{\lambda} \tau>1 $ and $d_n\to +\infty$ as $n\to +\infty$, we have \[ \lim_{n\to +\infty} I_{\bar{\lambda}, \bar{\mu}} (w_n)=-\infty. \] Hence, our claim is proved. It follows that $I_{\bar{\lambda},\bar{\mu}}$ has no global minimum. Therefore, by Lemma \ref{lem 2.1}(a), there exists a sequence $\{ u_n\}$ of critical points of $I_{\bar{\lambda},\bar{\mu}}$ such that $\lim_{n\to +\infty} \|u_n\|=+\infty$, and the proof is complete. \end{proof} \begin{remark}\label{r3.2}\rm Under the conditions $A=0$ and $B=+\infty$, from Theorem \ref{t3.1} we see that for every $\lambda>0$ and for each $\mu\in]0,\frac{1}{2G_\infty}[$, problem \eqref{1.1} admits a sequence of classical solutions which is unbounded in $X$. Moreover, if $G_\infty=0$, the result holds for every $\lambda>0$ and $\mu>0$. \end{remark} Here, we present a concrete application of Theorem \ref{t3.1}. \begin{example}\label{ex3.3}\rm Let $f:[0, 1]\times \mathbb{R}\to \mathbb{R}$ be a function defined by \begin{equation*} f(x,t)=\begin{cases} 0, & (x,t)\in [0,1]\times \{ 0\},\\ x^2t\left(2-2\sin(\ln |t|)-\cos(\ln |t|)\right), & (x,t)\in [0,1]\times(\mathbb{R}-\{0\}). \end{cases} \end{equation*} A direct calculation yields \begin{equation*} F(x,t)=\begin{cases} 0, & (x,t)\in [0,1]\times \{ 0\},\\ x^2t^2\big(1-\sin(\ln |t|)\big), & (x,t)\in [0 ,1]\times(\mathbb{R}-\{0\}). \end{cases} \end{equation*} It is easy to see that $A=0$ and $B=37/96$. Hence, denoting $u^+:=\max\{u,0\}$, by taking Remark \ref{r3.2} into account, we have that for every $(\lambda,\mu)\in]0,\frac{2^{10}\pi^4}{333}[\times]0,1[$ the problem \begin{gather*} u^{(iv)}(x)= \lambda f(x,u(x)) \quad \text{in } [0,1],\\ u(0)=u'(0)=0,\\ u''(1)=0,\quad u'''(1)=\mu\left(u^+(1)+1\right) \end{gather*} has a sequence of classical solutions which is unbounded in $X$. \end{example} The following corollary is a direct consequence of Theorem \ref{t3.1}. \begin{corollary}\label{cor3.3} Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be an $L^1$-Carat\'{e}odory function. Assume that hypothesis $\rm(A1)$ holds, and \[ A<\frac{1}{2},\quad B>\frac{32\pi^4}{27}. \] Then, for every continuous function $g:\mathbb{R}\to \mathbb{R}$, whose potential $G$ satisfying the conditions $\inf_{\xi>0}G(\xi)=0$ and \eqref{3'.1}, if we put \[ \mu_{G}:=\frac{1}{2G_\infty}\left(1-2A\right), \] where $\mu_{G}=+\infty$ when $G_\infty=0$, the problem \begin{gather*} u^{(iv)}(x)=f(x,u(x)) \quad \text{in } [0,1],\\ u(0)=u'(0)=0,\\ u''(1)=0,\quad u'''(1)=\mu g(u(1)) \end{gather*} has an unbounded sequence of classical solutions for every $\mu\in]0,\mu_{G}[$ in $X$. \end{corollary} We remark that Theorem \ref{t1.1} follows immediately from Theorem \ref{t3.1}. Now, we point out a special situation of our main result when the nonlinear term has separated variables. To be precise, let $h\in L^1([0,1])$ such that $h(x)\geq 0$ a.e. $x\in [0,1]$, $h\not\equiv 0,$ and let $k:\mathbb{R}\to\mathbb{R}$ be a nonnegative continuous function. Consider the fourth-order problem \begin{equation}\label{sep} \begin{gathered} u^{(iv)}(x)= \lambda h(x)k(u(x)) \quad \text{in } [0,1],\\ u(0)=u'(0)=0,\\ u''(1)=0,\quad u'''(1)=\mu g(u(1)). \end{gathered} \end{equation} Put $K(\xi):=\int_0^\xi k(t)dt$ for all $\xi\in\mathbb{R}$, and set $\|h\|_1:=\int_0^1 h(x)dx$ and $h_0:=\int_{\frac{3}{4}}^{1}h(x)dx$. \begin{corollary}\label{cor3.4} Suppose that $$ \liminf_{\xi\to+\infty}\frac{K(\xi)}{\xi^2} <\frac{27h_0}{64\pi^4\|h\|_1}\limsup_{\xi\to+\infty}\frac{K(\xi)}{\xi^2}. $$ Then, for each $$ \lambda\in\Big]\frac{32\pi^4}{(27h_0)\limsup_{\xi\to+\infty} \frac{K(\xi)}{\xi^2}},\frac{1}{(2\|h\|_1) \liminf_{\xi\to+\infty}\frac{K(\xi)}{\xi^2}}\Big[, $$ and every continuous function $g:\mathbb{R}\to \mathbb{R}$, whose potential $G$ satisfies the conditions $\inf_{\xi>0}G(\xi)=0$ and \eqref{3'.1}, if we put \[ \mu'_{G,\lambda}:=\frac{1}{2G_\infty}\Big(1-(2\lambda\|h\|_1) \liminf_{\xi\to+\infty}\frac{K(\xi)}{\xi^2}\Big), \] where $\mu'_{G,\lambda}=+\infty$ when $G_\infty=0$, problem \eqref{sep} has an unbounded sequence of classical solutions for every $\mu\in]0,\mu'_{G,\lambda}[$ in $X$. \end{corollary} Put \[ A':=\liminf_{\xi\to 0^+}\frac{\int_0^1 \max_{|t|\leq\xi}F(x,t)dx}{\xi^2},\quad B':=\limsup_{\xi\to 0^+}\frac{\int_{3/4}^1 F(x,\xi)dx}{\xi^2}. \] Using Lemma \ref{lem 2.1}(b) and arguing as in the proof of Theorem \ref{t3.1}, we can obtain the following multiplicity result. \begin{theorem}\label{t3.6} Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be an $L^1$-Carat\'{e}odory function. Assume that {\rm (A1)} and \begin{itemize} \item[(A3)] $A'<\frac{27}{64\pi^4} B'$, \end{itemize} are satisfied. Then, for every $\lambda\in\Lambda:=]\frac{32\pi^4}{27B'},\frac 1{2A'}[$ and for every continuous function $g:\mathbb{R}\to \mathbb{R}$, whose potential $G$ satisfying the conditions $\inf_{\xi>0}G(\xi)=0$ and \begin{equation*} G_0:=\limsup_{\xi\to 0^+}\frac{\max_{|t|\leq\xi}G(t)}{\xi^2}<+\infty, \end{equation*} if we put \[ \widetilde{\mu}_{G,\lambda}:=\frac{1}{2G_0} (1-2A'\lambda), \] where $\widetilde{\mu}_{G,\lambda}=+\infty$ when $G_0=0$, for every $\mu\in[0,\widetilde{\mu}_{G,\lambda})$ problem \eqref{1.1} has a sequence of classical solutions, which converges strongly to zero in $X$. \end{theorem} \begin{remark}\label{rem3.7}\rm Applying Theorem \ref{t3.6}, results similar to Theorem \ref{t1.1} and Corollaries \ref{cor3.3} and \ref{cor3.4} can be obtained. We omit the discussions here. \end{remark} Now, we put \[ A'':=\liminf_{\xi\to+\infty}\frac{\max_{|t|\leq\xi}G(t)}{\xi^2},\quad B'':=\limsup_{\xi\to+\infty}\frac{G(\xi)}{\xi^2}. \] By reversing the roles of $\lambda$ and $\mu$, we can obtain the following result. \begin{theorem}\label{t3.5} Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous function. Assume that \begin{itemize} \item[(A4)] $A''<\frac{27}{64 \pi^4}~B''$. \end{itemize} Then, for every $\mu\in\Gamma:=]\frac{32\pi^4}{27B''},\frac 1{2A''}[$ and for every $L^1$-Carat\'{e}odory function $f:[0,1]\times\mathbb{R}\to \mathbb{R}$, whose potential $F$ is a nonnegative function satisfying the condition \begin{equation}\label{e4.1} F_\infty:=\limsup_{\xi\to+\infty}\frac{\int_0^1 \max_{|t|\leq\xi}F(x,t)dx}{\xi^2}<+\infty, \end{equation} there exists $\lambda_{F,\mu}$, where \[ \lambda_{F,\mu}:=\frac{1}{2 F_\infty}(1-2A''\mu), \] such that for every $\lambda\in ]0,\lambda_{F,\mu}[$, problem \eqref{1.1} has an unbounded sequence of classical solutions in $X$. \end{theorem} \begin{proof} Fix $\bar{\mu}\in\Gamma$ and $f$ satisfying our assumptions. Since $\bar{\mu}<\frac{1}{2A''}$, we have $\lambda_{F,\bar{\mu}}>0$. Now fix $\bar{\lambda}\in \left]0,\lambda_{F,\bar{\mu}}\right[$. Set \begin{gather*} \widetilde{\Psi}_{\bar{\lambda},{\bar{\mu}}}(u) :=\frac{{\bar{\lambda}}}{{\bar{\mu}}}\int_0^1 F(x,u(x))dx+G(u(1)), \\ \widetilde{I}_{{\bar{\lambda}},{\bar{\mu}}}(u) :=\Phi(u)-{\bar{\mu}}\widetilde{\Psi}_{\bar{\lambda},{\bar{\mu}}}(u), \end{gather*} for all $u\in X$. Clearly, $\widetilde{I}_{{\bar{\lambda}},{\bar{\mu}}}={I}_{{\bar{\lambda}},{\bar{\mu}}}$. Let ${\xi_n}$ be a sequence of positive numbers such that $\lim_{n\to +\infty}\xi_n=+\infty$ and $$ \lim_{n\to+\infty}\frac{\max_{|t|\leq\xi_n}G(t)}{\xi_n^2}=A''. $$ Let $r_n={\xi_n^2}/{2}$ for all $n\in\mathbb{N}$. Arguing as in the proof of Theorem \ref{t3.1} and from the conditions (A4) and \eqref{e4.1} we obtain \[ \gamma\leq\liminf_{n\to+\infty} \varphi(r_n)\leq 2\frac{\bar{\lambda}}{\bar{\mu}}F_\infty+ 2 A''<+\infty. \] Therefore, from $\bar{\lambda} \in ]0,\lambda_{F,\bar{\mu}}[$ we obtain $\bar{\mu}<1/\gamma$. Let $\bar{\mu}$ be fixed. We claim that the functional $\widetilde{I}_{{\bar{\lambda}},{\bar{\mu}}}$ is unbounded from below. Since $1/\bar{\mu}<\frac{27 }{32 \pi^4} B''$, there exist a sequence $\{d_n\}$ and $\theta>0$ such that $\lim_{n\to +\infty}d_n= +\infty$ and \begin{equation}\label{e3.5} \frac {1}{\bar{\mu}}< \theta<\frac{27 }{32 \pi^4}\frac{G(d_n)}{d_n^2} \end{equation} for all $n\in\mathbb{N}$ large enough. Now, for every $n\in \mathbb{N}$, let $w_n\in X$ the function as given in \eqref{test}. Since $F$ is nonnegative, from \eqref{e3.5} we have \[ \widetilde{\Psi}_{\bar{\lambda},{\bar{\mu}}}(w_n)\geq G(d_n) > \frac{32}{27} \pi^4 \theta d_n^2. \] It follows that \[ \widetilde{I}_{{\bar{\lambda}},{\bar{\mu}}}(w_n) =\Phi(w_n)-{\bar{\mu}}\widetilde{\Psi}_{\bar{\lambda},{\bar{\mu}}}(w_n) \leq \frac{32}{27} \pi^4(1-\bar{\mu} \theta)d_n^2<0 \] for all $n\in\mathbb{N}$ large enough. Therefore, $\lim_{n\to +\infty} \widetilde{I}_{{\bar{\lambda}},{\bar{\mu}}}(w_n)=- \infty$, and the proof is complete. \end{proof} \begin{corollary}\label{c3.6} Assume that $g: \mathbb{R}\to \mathbb{R}$ be a nonpositive continuous function such that \[ \liminf_{\xi\to+\infty}\frac{-\int_0^\xi g(t)dt}{\xi^2}=0,\quad \limsup_{\xi\to+\infty}\frac{-\int_0^\xi g(t)dt}{\xi^2}=+\infty. \] Then, for each $\mu>0$ and for every nonnegative continuous function $f:\mathbb{R}\to\mathbb{R}$ satisfying the condition \[ f_\infty:=\limsup_{\xi\to+\infty}\frac{\int_0^\xi f(t) dt}{\xi^2} <+\infty, \] and for each $\lambda\in\big]0,\frac{1}{2 f_\infty}\big[$, problem \eqref{e3.6} admits infinitely many classical solutions. \end{corollary} \begin{thebibliography}{99} \bibitem{BR} S. Baraket, V. R\u{a}dulescu; {Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent}, {\it Adv. Nonlinear Stud.}, \textbf{16} (2016), 409--419. \bibitem{B} G. 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