MULTIPLICITY RESULTS FOR A NEUMANN BOUNDARY VALUE PROBLEM INVOLVING THE $P(X)$-LAPLACIAN

Abstract

In this paper we are interested in the multiplicity of weak solutions to the following Neumann problem involving the $p(x)$-Laplacian operator $$ \left\{ \begin{array}{ll} -\delta_{p(x)}u+\mid u \mid^{p(x)-2}u=\lambda \alpha (x)f(u)+\beta (x)g(u) \ \ \ &in \ \Omega \\ \frac{\partial u}{\partial v}=0 \ \ \ &on \ \Omega\end{array} \right . $$ We establish the existence of at least three solutions to this problem by using, as main tool, a recent variational principle due to Ricceri.