On eigenvalues problems for the $p(x)$-Laplacian

This paper studies nonlinear eigenvalues problems with a double non homogeneity governed by the $p(x)$-Laplacian operator, under the Dirichlet boundary condition on a bounded domain of ${\mathbb{R}}^N(N\ge 2)$. According to the type of the nonlinear part (sublinear, superlinear) we use the Lagrange multiplier’s method, the Ekeland’s variational principle and the Mountain-Pass theorem to show that the spectrum includes a continuous set of eigenvalues, which can in some contexts be all the set ${\mathbb{R}}_{+}^{\ast }$. Moreover, we show that the smallest eigenvalue obtained from the Lagrange multipliers is exactly the first eigenvalue in the Ljusternik-Schnirelman eigenvalues sequence. Key words: Nonlinear eigenvalue problems, $p(x)$-Laplacian, Lagrange multipliers, Ekeland variational principle, Ljusternik-Schnirelman principle, Mountain-Pass theorem.