On critical double phase problems in ${\mathbb{R}}^N$ involving variable exponents

Abstract

We establish a Lions-type concentration-compactness principle and its variant at infinity for Musielak-Orlicz-Sobolev spaces associated with a double phase operator with variable exponents. Based on these principles, we demonstrate the existence and concentration of solutions for a class of critical double phase equations of Schrödinger type in ${\mathbb{R}}^N$ involving variable exponents with various types of potentials. Our growth condition is more appropriately suited compared to the existing works.