Galerkin-type minimizers to a competing problem for (p, q)-Laplacian with variable exponents

Abstract

This study focuses on a sequence of approximate minimizers for the functional [J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,] where (\Omega\subset\mathbb{R}^N) ((N\geq 3)) is a bounded domain, and (p_i,q_i\in C(\overline{\Omega})) with (1\lt p_i,q_i\lt +\infty) for all (i \in {1,\ldots,N}). We establish the convergence result to the infimum of (J(u)) when (F:\mathbb{R}\to\mathbb{R}) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional.