Logarithmic double phase problems with generalized critical growth

TL;DR

This paper investigates logarithmic double phase problems with variable exponents and generalized critical growth, establishing new embedding results and a concentration compactness principle to prove multiplicity of solutions.

Contribution

It introduces novel embedding theorems and a concentration compactness principle for Musielak-Orlicz Sobolev spaces with logarithmic double phase structure, enabling multiplicity results.

Findings

Established new continuous and compact embedding results.

Proved a concentration compactness principle for Musielak-Orlicz Sobolev spaces.

Demonstrated multiplicity of solutions for superlinear and sublinear cases.

Abstract

In this paper we study logarithmic double phase problems with variable exponents involving nonlinearities that have generalized critical growth. We first prove new continuous and compact embedding results in order to guarantee the well-definedness by studying the Sobolev conjugate function of our generalized $N$-function. In the second part we prove the concentration compactness principle for Musielak-Orlicz Sobolev spaces having logarithmic double phase modular function structure. Based on this we are going to show multiplicity results for the problem under consideration for superlinear and sublinear growth, respectively.