Double phase anisotropic variational problems involving critical growth

TL;DR

This paper establishes existence and multiplicity results for double phase anisotropic variational problems with critical growth, introducing new concentration-compactness principles applicable even to variable exponent Laplace equations.

Contribution

It develops new concentration-compactness principles at infinity and applies them to prove existence and multiplicity of solutions for complex anisotropic problems with critical growth.

Findings

Existence of a nontrivial nonnegative solution.

Infinitely many solutions for symmetric nonlinearities.

Results extend to $p(\cdot)$-Laplace equations.

Abstract

In this paper, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. By employing these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the $p(\cdot)$-Laplace equations.