A fixed-point iteration for nonlinear PDE involving variable exponent function

Abstract

We study a nonlinear Dirichlet boundary value problem with the variable exponent p(x)-Laplacian and a reaction term. First, we prove the existence and uniqueness of a weak solution by minimizing the associated energy functional. To deal with possible singularities in the operator, we introduce a relaxed energy through a regularization method. We then show that this relaxed functional Γ-converges to the original one, and that the minimizers converge to the true solution. On this basis, we design a fixed point iteration scheme. We prove convergence result of the fixed-point procedure. The theoretical analysis is complemented by a finite element discretization and numerical experiments, which demonstrate the effectiveness, stability, and convergence properties of the proposed scheme.