Abstract
In this article, we are concerned with a class of nonlinear partial differential elliptic equations with Dirichlet boundary data. The key feature of this paper consists in competition effects of two generalized differential operators, which extend the standard operators with variable exponent. This class of problems is motivated by phenomena arising in non-Newtonian fluids or image reconstruction, which deal with operators and nonlinearities with variable exponents. We establish an existence property in the framework of small perturbations of the reaction term with indefinite potential. The mathematical analysis developed in this paper is based on the theory of anisotropic function spaces. Our analysis combines variational arguments with energy estimates.