Existence of Solutions to Anisotropic Elliptic Inequalities with Variable Exponents in Unbounded Domains

Abstract

The study addresses the problem of the existence of generalized solutions to anisotropic elliptic variational inequalities with variable nonlinearity exponents in unbounded domains. The primary objective is to establish solvability conditions for a class of inequalities containing lower-order terms with non-polynomial growth, thereby extending known results limited to isotropic cases or polynomial growth. The methodology is based on the application of the theory of pseudomonotone operators, functional analysis, and properties of anisotropic Lebesgue and Sobolev-Orlicz spaces with variable exponents. An existence theorem is proven for solutions to second-order anisotropic variational inequalities for operators that include higher-order terms with variable exponent power growth and lower-order terms with non-polynomial growth. The results are applicable to the qualitative theory of boundary value problems for quasilinear elliptic equations and can be used for the further development of the theory of inequalities in unbounded domains.