Remarks on Nonlocal Dirichlet Problems

Abstract

We study a nonlocal Dirichlet problem with the (p(b(u)),q(b(u)))-Laplacian operator and integrable data on a bounded domain with smooth boundary. We establish the existence of at least one weak solution in the case the variable exponents of the leading operator depend on the solution u, without assuming any growth conditions on g. The proof is based on the characterization of the energy functional associated to the problem, using the methods of the calculus of variations.