Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions (Q450600)

Given a smooth bounded open domain \(\Omega\) of \(\mathbb{R}^N\), the authors consider the following class of partial differential equations with Neumann boundary conditions NEWLINE\[NEWLINE(P_{\lambda,\mu})\qquad\begin{cases} -\Delta_{p(x)} u+\alpha(x)|u|^{p(x)-2} u=\lambda f(x,u)\quad &\text{in }\Omega,\\ |\nabla u(x)|^{p(x)-2} {\partial u\over\partial\nu}=\mu g(u)\quad &\text{on }\partial\Omega,\end{cases}NEWLINE\]NEWLINE where \(\Delta_{p(x)}u= \text{div}(|\nabla u|^{p(x)-2}\nabla u)\) is the \(p(x)\)-Laplacian operator, \(p\in C(\overline\Omega)\), \(p(x)> N\).NEWLINENEWLINE Under a condition on \(f\) that implies in particular that NEWLINE\[NEWLINEF(x,\xi)= \int^\xi_0 f(x,s)\,dsNEWLINE\]NEWLINE has some oscillating behavior, they prove that when \(\lambda\) belongs to some interval depending on \(f\), \(\alpha\), \(p\) and when \(\mu\) is small enough, \((P_{\lambda,\mu})\) has a sequence of weak solutions which is unbounded in \(W^{1, p(x)}(\Omega)\). These solutions are local minima of the action functional associated to the problem. To prove the existence of these local minima, the authors use an abstract result stated by \textit{G. Bonanno} and \textit{G. M. Bisci} [Bound. Value Probl. 2009, Article ID 670675. 20 p. (2009; Zbl 1177.34038)].