\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 68, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/68\hfil Second-order boundary-value problems] {Second-order boundary-value problems with variable exponents} \author[G. D'Agu\`i \hfil EJDE-2014/68\hfilneg] {Giuseppina D'Agu\`i} \address{Giuseppina D'Agu\`{\i} \newline Department of Civil, Information Technology, Construction Environmental Engineering and Applied Mathematics University of Messina, 98166 - Messina, Italy} \email{dagui@unime.it} \thanks{Submitted December 17, 2013. Published March 7, 2014.} \subjclass[2000]{34B15, 34L30} \keywords{Neumann problem; $p(x)$-Laplacian; variable exponent Sobolev spaces} \begin{abstract} In this article, we study ordinary differential equations with $p(x)$-Laplacian and subject to small perturbations of nonhomogeneous Neumann conditions. We establish the existence of an unbounded sequence of weak solutions by using variational methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we consider the following boundary value problem involving an ordinary differential equation with $p(x)$-Laplacian operator, and nonhomogeneous Neumann conditions: \begin{equation} \label{ePlm} \begin{gathered} -(|u'(x)|^{p(x)-2} u'(x))' + \alpha(x) |u(x)|^{p(x) - 2} u(x) =\lambda f(x,u) \quad \text{in }]0,1[ \\ | u'(0)|^{p(0)-2}u'(0) =-\mu g(u(0)), \\ | u'(1)|^{p(1)-2} u'(1)=\mu h(u(1)). \end{gathered} \end{equation} Here $p\in C({[0,1]}, \mathbb{R})$, $f: [0,1] \times \mathbb{R} \to \mathbb{R}$ is a Carath\'eodory function, (that is $x \to f(x,t)$ is measurable for all $t \in \mathbb{R}$, $t \to f(x,t)$ is continuous for almost every $x \in [0,1]$), $g,h:\mathbb{R}\to \mathbb{R}$ are nonnegative continuous functions, $\lambda$ and $\mu$ are real parameters with $\lambda>0$ and $\mu\geq 0$, $\alpha \in L^{\infty}([0,1])$, with $\operatorname{ess\,inf}_{[0,1]} \alpha >0$. The necessary framework for the study of problems involving the $p(x)$-Laplacian operator is represented by the functions spaces with variable exponent $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$. The basic properties of such spaces can be found in \cite{fan,kovacik}, and for a complete overview on this subject we refer to \cite{diening,musielak}. Differential problems with nonstandard $p(x)$-growth have been studied by many authors, see for instance \cite{cammaroto,Dasci,fandeng,xlfan-sgdeng,mihailescu,qian,yao} and the references therein. When $p(x)=p$ is constant, \eqref{ePlm} reduces to the ordinary $p$-Laplacian problem \begin{equation} \label{e1.1} \begin{gathered} -(|u'(x)|^{p-2} u'(x))' + \alpha(x) |u|^{p - 2} u =\lambda f(x,u) \quad \text{in } ]0,1[ \\ |u'(0)|^{p-2} u'(0) =-\mu g(u(0)), \\ |u'(1)|^{p-2} u'(1) =\mu h(u(1)). \end{gathered} \end{equation} Some results concerning such a problem, when $h\equiv g,$ can be found in \cite{BoMowi} (see for instance Theorem 4.1), where the authors obtain infinitely many solutions for a class of variational-hemivariational inequality by using the nonsmooth analysis. In \cite{faww}, the authors obtain one solution for weighed $p(x)$-Laplacian ordinary system, generalizing some results obtained by Hartman \cite{Hart} and Mawhin \cite{mawhin} which studied, respectively, the constant cases $p(x)=2$ and $p(x)=p$. Zhang \cite{Zhang}, via Leray-Schauder degree, obtained sufficient conditions for the existence of one solution for a weighted $p(x)$-Laplacian system boundary value problem. By using minimax methods, in \cite{WangYuan}, the authors study the periodic solutions for a class of systems with nonstandard $p(x)$-growth. In the present paper, under an appropriate oscillating behaviour of the primitive of the nonlinearity and a suitable growth at infinity of the primitives of $g$ and $h$, the existence of infinitely many weak solutions for \eqref{ePlm}, is obtained, for all $\lambda$ belonging to a precise interval and provided $\mu$ small enough (Theorem \ref{infinitesoluzioni}). We refer also to \cite{dagui,Bodagui} and the references therein for arguments closely related to our results. Here, as a particular case, we point out the following result on the existence of infinitely many solutions to problem $(P_{\lambda, \mu})$, when $\alpha(x)=1$ for all $x\in{[0,1]}$. \begin{theorem}\label{introduzione} Let $p \in C([0,1], \mathbb{R})$ such that $1< p^- := \min_{x \in [0,1]} p(x)\leq p^+ := \max_{x \in [0,1]} p(x)$ and let $f:\mathbb{R} \to \mathbb{R}$ be a nonnegative continuous function. Put $F(\xi)={ \int_{0}^{\xi}f(t)dt}$ for all $\xi\in\mathbb{R}$ and assume that $$ \liminf_{\xi \to +\infty} \frac{F(\xi)}{\xi^{p^-}} = 0 \quad \text{and} \quad \limsup_{\xi \to +\infty} \frac{F(\xi)}{\xi^{p^+}} = + \infty. $$ Then, for each $g:\mathbb{R}\to \mathbb{R}$ and for each $h:\mathbb{R}\to \mathbb{R}$ nonnegative continuous functions such that $$ \lim_{\xi\to +\infty}\frac{g(\xi)}{\xi^{p^{-}-1}} =\lim_{\xi\to +\infty}\frac{h(\xi)}{\xi^{p^{-}-1}}=0, $$ the problem \begin{gather*} -(|u'(x)|^{p(x)-2} u'(x))' + |u|^{p(x) - 2} u = f(u) \quad \text{in } ]0,1[ \\ | u'(0)|^{p(0)-2}u'(0) =-g(u(0)), \\ | u'(1)|^{p(1)-2} u'(1)=h(u(1)) \end{gather*} admits infinitely many distinct pairwise nonnegative weak solutions. \end{theorem} It is worth mentioning that in the study of existence of infinitely many solutions for the $p(x)$-Laplacian, symmetric assumptions (see \cite{yao}) or change sign hypothesis on the nonlinearity (see \cite{cammaroto}) are requested, while, in our main result such conditions are not required (see also Remark \ref{rem}). In particular, here, we can study problems with positive nonlinearity (see Example \ref{esempio}). This paper is arranged as follows. In Section \ref{paragrafo2}, some definitions and results on variable exponent Lebesgue and Sobolev spaces are collected. In particular, in Proposition \ref{prop}, an appropriate embedding constant of the space $W^{1,p(x)}([0,1])$ into $C^0([0,1])$ is estimated. Moreover, the abstract critical points theorem (Theorem \ref{thbona}) is recalled. Finally, in Section \ref{paragrafo3}, our main result is established, then some particular case and some example are presented. \section{Variable exponent Lebesgue and Sobolev space}\label{paragrafo2} Here and in the sequel, we assume that $p \in C([0,1],\mathbb{R})$ satisfies the condition \begin{equation} \label{funzionep} 1< p^- := \min_{x \in [0,1]} p(x)\leq p^+ := \max_{x \in [0,1]} p(x). \end{equation} The variable exponent Lebesgue spaces are defined as follows \[ L^{p(x)}([0,1]) = \big\{ u :[0,1] \to \mathbb{R} : u \text{ is measurable and }\int_{0}^1 |u|^{p(x)} dx < + \infty \big\}. \] On $L^{p(x)}([0,1])$, we consider the norm \[ \|u\|_{L^{p(x)}([0,1])}:= \inf\big\{ \lambda > 0 : \int_{\Omega} |\frac{u(x)}{\lambda}|^{p(x)} dx \leq 1\big\}. \] Let $X$ be the generalized Lebesgue-Sobolev space $W^{1,p(x)}([0,1])$ defined by \[ W^{1,p(x)}([0,1]) := \big\{u : u \in L^{p(x)}([0,1]) , u' \in L^{p(x)}([0,1]) \big\}, \] endowed with the norm \begin{equation} \label{norma} \|u\|_{W^{1,p(x)}([0,1])} := \|u\|_{L^{p(x)}([0,1])} + \| | u| \|_{L^{p(x)}([0,1])}. \end{equation} It is well known (see \cite{fan}) that, in view of \eqref{funzionep}, both $L^{p(x)}([0,1])$ and $W^{1,p(x)}([0,1])$, with the respective norms, are separable, reflexive and uniformly convex Banach spaces. Moreover, since $\alpha \in L^{\infty}([0,1])$, with $\alpha_- := \operatorname{ess\,inf}_{x \in [0,1]}\alpha(x) > 0$ is assumed, the norm \[ \|u\|_{\alpha}:= \inf \big\{ \sigma > 0 : \int_{0}^1 \Big(\ |\frac{ u'(x)}{\sigma}|^{p(x)} + \alpha(x) |\frac{u(x)}{\sigma}|^{p(x)} \Big)\, dx \leq 1\big\}, \] on $W^{1,p(x)}([0,1])$ is equivalent to that introduced in \eqref{norma}. Next, we give an estimate on the embedding constant $m$ of $W^{1,p(x)}([0,1])$ with norm $\|\cdot\|_\alpha$ in $C^0([0,1])$. \begin{proposition}\label{prop} For all $u\in W^{1,p(x)}([0,1])$, one has \begin{equation} \label{immersione} {\|u\|_{C^0([0,1]} \leq m \|u\|_{\alpha},} \end{equation} where $$ m= \begin{cases} 2\Big[\frac{1}{\alpha_-^{\frac{p^+}{p^-(1-p^+)}}+1}\Big]^{1/p^+} +\Big[ 1-\frac{1}{\alpha_-^{\frac{p^+}{p^-(1-p^+)}}+1} \Big]^{1/p^+} \frac{2}{\alpha_-^{1/p^-}} & \text{if } \alpha_-< 1\\[6pt] 2\Big[\frac{1}{\alpha_-^{\frac{1}{1-p^+}}+1}\Big]^{1/p^+} +\Big[ 1-\frac{1}{\alpha_-^{\frac{1}{(1-p^+)}}+1} \Big]^{1/p^+}\frac{2}{\alpha_- ^{1/p^+}} & \text{if}\,\, \alpha_-\geq 1.\\ \end{cases} $$ \end{proposition} \begin{proof} First we observe that $$ |u(t)|\leq \int_0^1 |u'(t)|dt+ \int_0^1 |u(t)|dt,\quad\forall u\in W^{1,p(x)}[(0,1)]. $$ Moreover, taking into account H\"older inequality in variable exponent Lebesgue space (see, for instance, \cite[Lemma 3.2.20]{diening}), one has \begin{gather*} \|u\|_{L^1{[0,1]}}\leq 2\|u\|_{L^{p(x)}{[0,1]}},\\ \|u'\|_{L^1{[0,1]}}\leq 2\|u'\|_{L^{p(x)}{[0,1]}}. \end{gather*} Therefore, \begin{equation}\label{immersione1} \|u(t)\|_{C^0([0,1])}\leq 2\|u\|_{W^{p(x)}{[0,1]}},\quad \forall u\in W^{1,p(x)}[(0,1)]. \end{equation} In the variable exponent Sobolev space, we consider the equivalent norm \begin{align*} \|u\|_\alpha &:=\inf\big\{ \lambda >0: \int_0^1 \Big(|\frac{u'(x)}{\lambda}|^{p(x)}+\alpha(x)|\frac{u(x)}{\lambda}|^{p(x)}\Big)dx \leq 1\big\}\\ &=\inf \{ \lambda >0:\rho_\alpha (\frac{u}{\lambda})\leq 1\}. \end{align*} From definition of $\|u\|_{\alpha}$ one has \begin{align*} 1&\geq\rho_\alpha\big(\frac{u}{\|u\|_\alpha}\big)\\ &=\int_0^1 \Big(|\frac{u'(x)}{\|u\|_\alpha}|^{p(x)}+\alpha(x)| \frac{u(x)}{\|u\|_\alpha}|^{p(x)}\Big)dx \\ &\geq \int_0^1 \Big(|\frac{u'(x)}{\|u\|_\alpha}|^{p(x)} +\alpha_-|\frac{u(x)}{\|u\|_\alpha}|^{p(x)}\Big)dx. \end{align*} Now we suppose that $\alpha_-< 1$, one has \[ 1\geq\int_0^1 \Big(|\frac{u'(x)}{\|u\|_\alpha}|^{p(x)} +|\frac{u(x)}{(\frac{1}{\alpha^-})^{1/p^-}\|u\|_\alpha}|^{p(x)}\Big)dx. \] This leads to \begin{gather} \int_0^1 |\frac{u'(x)}{\|u\|_\alpha}|^{p(x)}dx=k\leq 1 \label{e2.5}\\ \int_0^1|\frac{u(x)}{\left(\frac{1}{\alpha^-}\right)^{1/p^-}\|u\|_a}|^{p(x)}dx = 1-k \leq 1. \label{e2.6} \end{gather} From \eqref{e2.5} and \eqref{e2.6}, dividing by respectively by $k$ and $1-k$, we obtain \begin{gather*} \|| u'|\|_{L^{p(x)}}\leq k^{1/p^+}\|u\|_\alpha, \\ \|| u|\|_{L^{p(x)}}\leq \frac{(1-k)^{1/p^+}}{\alpha_-^{1/p^-}}\|u\|_\alpha. \end{gather*} Therefore, \begin{align*} \|| u|\|_{W^{p(x)}} &\leq k^{1/p^+}\|u\|_\alpha +\frac{(1-k)^{1/p^+}}{\alpha_-^{1/p^-}}\|u\|_\alpha = \Big(k^{1/p^+}+\frac{(1-k)^{1/p^+}}{\alpha_-^{1/p^-}}\Big)\|u\|_\alpha \\ &\leq \Big\{ \Big[\frac{1}{\alpha_-^{\frac{p^+}{p^-(1-p^+)}}+1}\Big]^{1/p^+} +\Big[ 1-\frac{1}{\alpha_-^{\frac{p^+}{p^-(1-p^+)}}+1} \Big]^{1/p^+} \frac{1}{\alpha_-^{1/p^-}} \Big\}\|u\|_\alpha. \end{align*} In a similar way, we work when $\alpha_-\geq 1$ and we obtain $$ \|| u|\|_{W^{p(x)}} \leq \Big\{\frac{1}{\big(\alpha_-^{\frac{1}{1-p^+}}+1\big)^{1/p^+}} +\big[ 1-\frac{1}{\alpha_-^{\frac{1}{(1-p^+)}}+1} \big]^{1/p^+}\frac{1}{\alpha^{1/p^+}} \Big\}\|u\|_\alpha. $$ Now, taking also into account \eqref{immersione1}, we claim the thesis. \end{proof} \begin{remark} {\rm It is worth mentioning that if $\alpha_-\geq 1$, the constant $m$ does not exceed $2$. Instead, when $\alpha_-< 1$, $m$ depend on $\alpha_-$ and in particular is less than $2(1+\frac{1}{\alpha_-})$.} \end{remark} In the sequel, $f:[0,1]\times\mathbb{R} \to \mathbb{R}$ is an $L^1$-Carath\'{e}odory function, $g,h:\mathbb{R}\to \mathbb{R}$ are two nonnegative continuous functions, and $\lambda$ and $\mu$ are real parameters. We recall that $f:[0,1]\times\mathbb{R} \to \mathbb{R}$ is an $L^1$-Carath\'{e}odory function if: \begin{itemize} \item[(1)] $x\mapsto f(x,\xi)$ is measurable for every $\xi \in \mathbb{R}$; \item[(2)] $\xi \mapsto f(x,\xi)$ is continuous for almost every $x \in [0,1]$; \item[(3)] for every $s>0$ there is a function $l_s \in L^1([0,1])$ such that $$ \sup_{|\xi|\leq s}|f(x, \xi)|\leq l_s(x) $$ for a.e. $x\in [0,1]$. \end{itemize} Put \begin{gather*} F(x,t)={ \int_{0}^{t}f(x,\xi)d\xi} \quad \text{for all } (x,t) \in [0,1]\times \mathbb{R}, \\ G(t)={ \int_{0}^{t}g(\xi)d\xi} \quad \text{for all } t \in \mathbb{R}, \\ H(t)={ \int_{0}^{t}h(\xi)d\xi} \quad \text{for all } t \in \mathbb{R}. \end{gather*} We recall that $u : [0,1] \to \mathbb{R}$ is a weak solution of problem \eqref{ePlm} if $u \in W^{1,p(x)}([0,1])$ satisfies the condition \begin{align*} &\int_{0}^1 | u'(x)|^{p(x) - 2} u'(x) v'(x) dx + \int_{0}^1 \alpha(x)|u(x)|^{p(x) - 2} u(x) v(x) dx \\ &-\lambda \int_{0}^1 f(x,u(x)) v(x) dx -\mu[g(u(0))v(0)+h(u(1))v(1)]= 0, \end{align*} for all $v \in W^{1,p(x)}([0,1])$. To prove our main theorem, we use critical point theory and in particular \cite[Theorem 2.1]{BoMo}, that we recall here. Let $X$ be a reflexive real Banach space, $\Phi: X \to \mathbb{R}$ is a (strongly) continuous, coercive, sequentially weakly lower semicontinuous and G\^{a}teaux differentiable function, $\Psi : X \to \mathbb{R}$ is a sequentially weakly upper semicontinuous and G\^{a}teaux differentiable function. For every $r > \inf_{X} \Phi$, put \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)}, \\ \overline{\gamma} := \liminf_{r\to +\infty}\varphi(r), \quad \delta := \liminf_{r\to (\inf_{X} \Phi)^+}\varphi(r). \end{gather*} \begin{theorem}\label{thbona} Under the above assumptions of $X$, $\Phi$ and $\Psi$, the following alternatives hold: \begin{itemize} \item[(a)] for every $r > \inf_{X} \Phi$ and every $\lambda \in ]0, \frac{1}{\varphi(r)}[$, the restriction of the functional $\Phi-\lambda\Psi$ to $\Phi^{-1}(]-\infty,r[)$ admits a global minimum, which is a critical point (local minimum) of $\Phi-\lambda\Psi$ in $X$. \item[(b)] if $\overline{\gamma} < +\infty$ then, for each $ \lambda \in ]0, \frac{1}{\overline{\gamma}}[$, the following alternative holds: either the functional $\Phi-\lambda\Psi$ has a global minimum, or there exists a sequence $\{u_n\}$ of critical points (local minima) of $\Phi-\lambda\Psi$ such that $\lim_{n\to +\infty}\Phi(u_n)=+\infty$. \item[(c)]If $\delta < +\infty$ then, for each $ \lambda \in ]0, \frac{1}{\delta}[$, the following alternative holds: either there exists a global minimum of $\Phi$ which is a local minimum of $\Phi-\lambda\Psi$, or there exists a sequence $\{u_n\}$ of pairwise distinct critical points (local minima) of $\Phi-\lambda\Psi$, with $\lim_{n\to +\infty}\Phi(u_n)= \inf_{X} \Phi$, which weakly converges to a global minimum of $\Phi$. \end{itemize} \end{theorem} \section{Main Result}\label{paragrafo3} In this section, we establish an existence result of infinitely many solutions to problem \eqref{ePlm}. Put $$ A := \liminf_{\xi \to + \infty} \frac{\int_{0}^1 \max_{| t |< \xi}F(x,t) dx}{\xi^{p^-}},\quad B : = \limsup_{\xi \to + \infty} \frac{\int_{0}^1 F(x,\xi) dx}{\xi^{p^+}}, $$ and \begin{equation}\label{lamda} \lambda_1= \frac{ \| \alpha \|_1}{p^- B}, \quad \lambda_2=\frac{1}{p^+ m^{p^-}A}, \end{equation} where $ \| \alpha \|_1$ is the usual norm in $L^1(\Omega)$ and $m$ is given by Proposition \ref{prop}. \begin{theorem}\label{infinitesoluzioni} Let $f:[0,1]\times\mathbb{R} \to \mathbb{R}$ an $L^1$-Carath\'{e}odory function. Assume that $$ \liminf_{\xi \to + \infty} \frac{\int_{0}^1 \max_{| t |< \xi}F(x,t) dx}{\xi^{p^-}} < \frac{p^-}{p^+ m^{p^-} \| \alpha \|_1} \limsup_{\xi \to + \infty} \frac{\int_{0}^1 F(x,\xi) dx}{\xi^{p^+}}. $$ Then, for each $\lambda \in \left]\lambda_1, \lambda_2 \right[$, for each $g:\mathbb{R}\to \mathbb{R}$ and for each $h:\mathbb{R}\to \mathbb{R}$ nonnegative continuous functions such that $$ G_{\infty}=\limsup_{\xi\to +\infty}\frac{G(\xi)}{\xi^{p^{-}}}< +\infty,\quad H_{\infty}=\limsup_{\xi\to +\infty}\frac{H(\xi)}{\xi^{p^{-}}}< +\infty, $$ and for each $\mu\in[0, \delta[$, with $$ \delta=\frac{1-m^{p^{-}}p^+ \lambda A}{m^{p^{-}}p^+ [G_{\infty}+H_{\infty}]}, $$ problem \eqref{ePlm} admits a sequence of weak solutions which is unbounded in the space $W^{1,p(x)}([0,1])$. \end{theorem} \begin{proof} Our aim is to apply Theorem \ref{thbona}. To this end, fix $\lambda, \mu$, $g$ and $h$ satisfying our assumptions. Let $X$ be the Sobolev space $W^{1,p(x)}([0,1])$. For any $u\in X$, set \begin{gather*} \Phi(u) := \int_{0}^1 \frac{1}{p(x)} \Big( |u'|^{p(x)} + \alpha(x) |u|^{p(x)} \Big) dx, \\ \Psi(u):= \int_{0}^1 F(x,u(x))dx+\frac{\mu}{\lambda}[G(u(0))+H(u(1))]. \end{gather*} It is well known that they satisfy all regularity assumptions requested in Theorem \ref{thbona} and that the critical points in $X$ of the functional $I_{\lambda} = \Phi - \lambda \Psi$ are precisely the weak solutions of problem \eqref{ePlm}. Let $\{ c_n\}$ be a real sequence of positive numbers such that $\lim_{n \to +\infty} c_n = + \infty$, and $$ \lim_{n \to + \infty} \frac{\int_{0}^1 \max_{| t |< c_n}F(x,t) dx}{c_n^{p^-}} = A. $$ Put $r_n = \frac{1}{p^+} \frac{c_n^{p^-}}{m^{p^-}}$, for each $n \in \mathbb{N}$ and $\Phi(v)< r_n$, then, owing to \cite[Proposition 2.2]{cammaroto}, one has $$ \| v \|_{\alpha} \leq \max \{ (p^+ r_n)^{\frac{1}{p^+}}, (p^+ r_n)^{\frac{1}{p^-}}\} = \frac{c_n}{m}, $$ and so, by \eqref{immersione}, $$ \max_{x \in [0,1]} | v(x)| \leq m \| v\|_{\alpha} \leq c_n. $$ Therefore, one has \begin{align*} \varphi(r_n) &\le \frac{\sup_{v\in \Phi^{-1}(]-\infty,r_n[)}\Psi(v)}{r_n} \\ &\le \frac{\int_{0}^1\max_{|t|\leq c_n}F(x,t)dx+\frac{\mu}{\lambda} \max_{|t|\leq c_n}[G(t)+H(t)]}{\frac{1}{p^+} \frac{c_n^{p^-}}{m^{p^-}}}\\ &\leq p^+ m^{p^-}\frac{\int_{0}^1\max_{|t| \leq c_n} F(x,t)dx+\frac{\mu}{\lambda}[G(c_n)+H(c_n)]}{c_n^{p^-}}, \quad \text{for all } n \in \mathbb{N}. \end{align*} Then $$ \overline{\gamma} \leq \liminf_{n \to + \infty} \varphi(r_n) \leq p^+ m^{p^-} A+ \frac{\mu}{\lambda}p^+ m^{p^-}[G_{\infty}+H_{\infty}] < + \infty. $$ Now, let $\{ \eta_n \}$ be a real sequence of positive numbers such that $\lim_{n \to +\infty} \eta_n = + \infty$, and \begin{equation}\label{B} \lim_{n \to + \infty} \frac{\int_{0}^1 F(x,\eta_n) dx}{\eta_n^{p^+}} = B. \end{equation} For each $n \in \mathbb{N}$, put $w_n(x)=\eta_n $, for all $x \in [0,1]$. Clearly $w_n (x) \in W^{1,p(x)}([0,1])$ for each $n \in \mathbb{N}$. Hence, we have \begin{align*} \Phi(w_n) & = \int_{0}^1 \frac{1}{p(x)} \Big( | w'_n|^{p(x)} + \alpha(x) |w_n|^{p(x)} \Big) dx \\ & = \int_{0}^{1} \frac{1}{p(x)} \alpha(x) \eta_n^{p(x)} dx \\ &\leq \int_{0}^{1} \frac{1}{p^-} \alpha(x) \eta_n^{p^+} dx = \frac{\eta_n^{p^+}}{p^-} \|\alpha\|_1. \end{align*} Now, for each $n \in \mathbb{N}$, one has \begin{align*} \Psi(w_n) &= \int_{0}^{1} F(x,w_n(x))dx +\frac{\mu}{\lambda}[G(w_n)+H(w_n)]\\ &= \int_{0}^{1} F(x,\eta_n)dx+\frac{\mu}{\lambda}[G(\eta_n)+H(\eta_n)], \end{align*} and so \begin{align*} I_{\lambda} (w_n) &= \Phi(w_n) - \lambda \Psi(w_n) \\ &\leq \frac{\eta_n^{p^+}}{p^-} \|\alpha\|_1 - \lambda\Big[ \int_{0}^{1} F(x,\eta_n)dx+\frac{\mu}{\lambda}[G(\eta_n) +H(\eta_n)]\Big]. \end{align*} Now, consider the following cases. If $B < +\infty$, we let $\epsilon \in ] 0, B - \frac{ \| \alpha \|_1}{\lambda p^-}[$. From \eqref{B}, there exists $\nu_{\epsilon}$ such that $$ \int_{0}^{1} F(x,\eta_n)dx > (B - \epsilon) \eta_n^{p^+}, \quad \text{for all } n > \nu_{\epsilon}, $$ and so \begin{align*} I_{\lambda} (w_n) &< \frac{\eta_n^{p^+}}{p^-} \| \alpha \|_1 - \lambda\big[ (B - \epsilon) \eta_n^{p^+} +\frac{\mu}{\lambda}[G(\eta_n)+H(\eta_n)]\big]\\ & = \eta_n^{p^+} \big[\frac{\|\alpha\|_1}{p^-} - \lambda (B - \epsilon) \big]-\mu [G(\eta_n)+H(\eta_n)]. \end{align*} Since $\frac{\|\alpha\|_1}{p^-} - \lambda (B - \epsilon) < 0$, one has $$ \lim_{n \to + \infty} I_{\lambda} (w_n) = - \infty. $$ If $B = +\infty$, fix $M > \frac{ \| \alpha \|_1}{\lambda p^-}$. From \eqref{B}, there exists $\nu_M$ such that $$ \int_{0}^{1} F(x,\eta_n)dx > M \eta_n^{p^+}, \quad \text{for all } n > \nu_M; $$ moreover \begin{align*} I_{\lambda} (w_n) &< \frac{\eta_n^{p^+}}{p^-} \|\alpha\|_1 - \lambda [M \eta_n^{p^+} +\frac{\mu}{\lambda}[G(\eta_n)+H(\eta_n)]] \\ &= \eta_n^{p^+} \big(\frac{\|\alpha\|_1}{p^-} - \lambda M \big) -\mu [G(\eta_n)+H(\eta_n)]. \end{align*} Since $\frac{\|\alpha\|_1}{p^-} - \lambda M < 0$, this leads to $$ \lim_{n \to + \infty} I_{\lambda} (w_n) = - \infty. $$ Taking into account that $$ ]\frac{\|\alpha\|_1}{{p^-}B}, \frac{1}{p^+ m^{p^-}A} [ \subseteq ] 0, \frac{1}{\overline{\gamma}}[, $$ and that $I_{\lambda}$ does not possess a global minimum, from part (b) of Theorem \ref{thbona}, there exists an unbounded sequence $\{u_n\}$ of critical points, and our conclusion is achieved. \end{proof} As an immediate consequence, here we present an existence result for the homogeneous Neumann problem \begin{equation} \label{ePl} %\tag{$P_{\lambda}$} \begin{gathered} -(|u'(x)|^{p(x)-2} u'(x))' + \alpha(x) |u|^{p(x) - 2} u =\lambda f(x,u) \quad \text{in } ]0,1[ \\ u'(0)=u'(1)=0. \quad \end{gathered} \end{equation} \begin{theorem}\label{omogeneo} Let $f:[0,1]\times\mathbb{R} \to \mathbb{R}$ an $L^1$-Carath\'{e}odory function. Assume that $$ \liminf_{\xi \to + \infty} \frac{\int_{0}^{1} \max_{| t |< \xi}F(x,t) dx}{\xi^{p^-}} < \frac{p^-}{p^+ m^{p^-} \| \alpha \|_1} \limsup_{\xi \to + \infty} \frac{\int_{0}^{1} F(x,\xi) dx}{\xi^{p^+}}. $$ Then, for each $\lambda \in ]\lambda_1, \lambda_2 [$, where $\lambda_1$ and $\lambda_2$ are given in \eqref{lamda}, problem \eqref{ePl} admits a sequence of weak solutions which is unbounded in $W^{1,p(x)}([0,1])$. \end{theorem} \begin{example}\label{esempio} {\rm Let $p\in C({[0,1]})$ satisfying \eqref{funzionep} and with $p^- \geq2$, and let $\{b_n\}_{n\in\mathbb{N}}$ and $\{a_n\}_{n\in\mathbb{N}}$ be the sequences defined as follows $b_1=2$, $b_{n+1}=(b_n)^{2(p^+ +1)}$ and $a_n=(b_n)^{2p^+}$ for all $n\in\mathbb{N}$. Moreover let $f:\mathbb{R}\to\mathbb{R}$ be a positive continuous function defined by $$ f(t) = \begin{cases} 2^{(p^+ +1)}\sqrt{1-(1-t)^2}+1 & t\in[0,2], \\[5pt] (a_n-(b_n)^{p^++1})\sqrt{1-(a_n -1-t)^2}+1 & t\in \cup_{n=1}^{+\infty}[a_n-2,a_n], \\[5pt] ((b_{n+1})^{p^++1}-a_n)\sqrt{1-(b_{n +1}-1-t)^2}+1 & t\in \cup_{n=1}^{+\infty}[b_{n+1}-2,b_{n+1}] ,\\[5pt] 1 &\text{otherwise.} \end{cases} $$ Put $ F(\xi)=\int_0^{\xi}f(t)dt$ for all $\xi\in\mathbb{R}$. In particular, one has $F(a_n)=a_n\frac{\pi}{2}+a_n$ for all $n\in\mathbb{N}$ and $F(b_n)=(b_n)^{p^+ +1}\frac{\pi}{2}+b_n$ for all $n\in\mathbb{N}$. Hence, $$ \liminf_{\xi\to +\infty}\frac{F(\xi)}{\xi^{p^-}} =\lim_{n\to +\infty}\frac{F(a_n)}{a_n^{p^-}}=0, $$ and $$ \limsup_{\xi\to +\infty}\frac{F(\xi)}{\xi^{p^+}} =\lim_{n\to +\infty}\frac{F(b_n)}{b_n^{p^+}}=+\infty. $$ Then, owing to Theorem \ref{infinitesoluzioni}, the problem \begin{gather*} -|u'|^{p(x)-2} u' + |u|^{p(x) - 2} u = f(u) \quad \text{in } ]0,1[\\ | u'(0)|^{p(0)-2}u'(0) =-\frac{1}{1+ (u(0))^2}, \\ | u'(1)|^{p(1)-2}u'(1) =u(1)\arctan u(1), \end{gather*} admits infinitely many weak solutions. }\end{example} \begin{remark}\label{rem} {\rm In \cite{yao} the existence of infinitely many solutions to problem \eqref{ePlm} when $\alpha(x) = 1$, is proved. Two of key assumptions of \cite[Theorem 4.8]{yao} are \begin{gather}\label{simmetriaf} f(x, -u) = -f(x,u), \quad \text{for all } x \in [0,1], \; u \in \mathbb{R}. \\ \label{simmetriag} g(-u) = -g(u), \quad \text{for all } u \in \mathbb{R}. \end{gather} Clearly, \cite[Theorem 4.8]{yao} cannot be applied to the problem of Example \ref{esempio}, since, there, the nonlinearity $f$ and the function $g$ are not symmetric for which \eqref{simmetriaf} and \eqref{simmetriag} are not satisfied.} \end{remark} \subsection*{Acknowledgements} The author was partially supported by Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`{a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) -- project ``Analisi non lineare e problemi ellittici''. \begin{thebibliography}{00} \bibitem{dagui} G. Bonanno, G. D'Agu\`{i}; \emph{On the Neumann problem for elliptic equations involving the $p$-Laplacian}, J. Math. Anal. 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