Remarks on Ricceri’s variational principle and applications to the p(x)p(x)-Laplacian equations

In this paper we give some remarks on a variational principle of Ricceri in the case of sequentially weakly lower semi-continuous functionals defined on a reflexive real Banach space. In particular we introduce the notions of Ricceri block and Ricceri box which are more convenient in some applications than the weakly connected components. Using the variational principle of Ricceri and a local mountain pass lemma, we study the multiplicity of solutions of the p(x)p(x)-Laplacian equations with Neumann, Dirichlet or no-flux boundary condition, and under appropriate hypotheses, in which the integral functionals need not satisfy the (PS) condition on the global space, we prove that the problem has at least seven solutions.