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| We study the existence of at least three distinct solutions for the following Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian:
\begin{equation}\tag{${\it N}_{\lambda}$}\label{eq:dir}
\left\{\begin{array}{l}
-\Delta _{p(x)}u + a(x) |u|^{p(x)-2}u\in \lambda\partial F(x,u)\;\; {\rm in}\;\;\Omega\\
\\
\displaystyle\frac{\partial u}{\partial \nu}=0\;\; {\rm on} \; \partial\Omega
\end{array}
\right.
\end{equation}
where $\Omega\subset {{\rm l}\kern-.17em{\rm R}} ^{N}$ is an open bounded domain with smooth boundary $\partial\Omega$, $a$ is a suitable function belonging to $L^{\infty}(\Omega)$, $\Delta _{p(x)}u :={\rm div} (|\nabla u|^{p(x)-2}\nabla u)$ denotes the $p(x)$-Laplace operator related to a convenient function $p$ of $C(\bar{\Omega})$, $\nu$ is the outward unit normal to $\partial\Omega$, $\lambda$ is a positive parameter and $\partial F(x,\xi)$ is the generalized gradient with respect to $\xi$ of a fixed function $F$ defined on $\Omega\times{{\rm l}\kern-.17em{\rm R}}$. The results are obtained by using a multiple critical points theorem for locally Lipschitz continuous functionals. |
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