Existence and multiplicity results for elliptic problems with Nonlinear Boundary Conditions and variable exponents

Abstract

By applaying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:\begin{equation*}\begin{gathered}-div[a(x, \nabla u)]+|u|^{p(x)-2}u=\lambda f(x,u), \quad \text{in }\Omega, \\a(x, \nabla u).\nu=\mu g(x,u), \quad \text{on } \partial\Omega,\end{gathered}\end{equation*}where $\lambda$, $\mu \in \mathbb{R}^{+},$$\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain ofsmooth boundary $\partial\Omega$ and $\nu$ is the outward normalvector on $\partial\Omega$. $p: \overline{\Omega} \mapsto\mathbb{R}$, $a: \overline{\Omega}\times \mathbb{R}^{N} \mapsto\mathbb{R}^{N},$ $f: \Omega\times\mathbb{R} \mapsto \mathbb{R}$and $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ arefulfilling appropriate conditions.