Existence of weak solutions for a nonlinear problem involving p(.)-Laplacian operator with mixed boundary conditions

In this paper, we consider a mixed boundary value problem to a class of quasi-linear elliptic operators containing p(.)-Laplacian operator. More precisely we consider the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least one non-trivial weak solution under some hypotheses and the existence of infinitely many weak solutions under some hypotheses.