Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent

Abstract

We establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic equations involving the anisotropic p ! ðÁÞ-Laplace operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev spaces and our main tool is the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz.

Solutions of an anisotropic nonlocal problem involving variable exponent

“…The differential operator involved in this article can be regarded as an extension of the isotropic p. /-Laplacian operator to anisotropic case,that is, to the E p. /-Laplacian operator which is a very general elliptic operator [7,12,20]. Such kind of operators can be seen as candidates for modelling phenomena which ask for distinct behavior of partial differential derivatives in various directions.…”

Dirichlet Problem for Stationary Anisotropic Higher-Order Partial Integrodifferential Equations with Variable Exponents of Nonlinearity

“…Among the works devoted to this circle of problems, we can especially mention [2,3,7,8,[13][14][15][16]. We do not know works dealing with nonlinear integrodifferential equations with constant exponents of nonlinearity.…”

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

“…In recent years many authors considered differential problems, in the context of the calculus of variations and of partial differential equations of elliptic and parabolic type, under general p, q−growth conditions and, as a special relevant case, related to p (x) −growth; i.e., variable exponents. Among them Vicenţiu D. Rȃdulescu who studied, in this framework of general growth, multiplicity of solutions for some nonlinear problems, qualitative analysis, anisotropic elliptic equations, eigenvalue problems and several other related questions; see for instance [20], [21], [5], [23], [2]. We like to explicitly dedicate this manuscript to Vicenţiu D. Rȃdulescu, with esteem and sympathy.…”

“…Theorem 1. Let p > 1 and u ∈ W 1,p loc (Ω) be a local minimizer of the energy integral (2) under the growth assumptions (3), (5). Then u is locally Lipschitz continuous in Ω.…”