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

Résumé

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 of
smooth boundary $\partial\Omega$ and $\nu$ is the outward normal
vector 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 $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ are
fulfilling appropriate conditions.