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

TL;DR

This paper develops a concentration-compactness principle for Musielak-Orlicz-Sobolev spaces with variable exponents, proving existence and concentration of solutions for critical double phase Schrödinger equations in R^N.

Contribution

It introduces a new Lions-type concentration-compactness principle tailored for variable exponent double phase problems, enabling analysis of solution existence and concentration.

Findings

Established a Lions-type concentration-compactness principle for Musielak-Orlicz-Sobolev spaces.

Proved existence of solutions for critical double phase Schrödinger equations with variable exponents.

Demonstrated solution concentration phenomena under new growth conditions.

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 Schr\"odinger 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.