Double phase problems with variable exponents depending on the solution and the gradient in the whole space $\mathbb{R}^N$

TL;DR

This paper develops new embedding results for Musielak-Orlicz Sobolev spaces with variable exponents depending on solutions and gradients, and proves existence and multiplicity of solutions for related double phase problems in unbounded domains.

Contribution

It introduces a novel class of Musielak-Orlicz Sobolev spaces with variable exponents and establishes key embedding theorems to analyze double phase problems in the whole space.

Findings

Established continuous and compact embeddings for the new Sobolev spaces.

Proved existence and multiplicity of weak solutions for the double phase problems.

Extended previous work by applying abstract critical point theory to these spaces.

Abstract

In this paper, we establish continuous and compact embeddings for a new class of Musielak-Orlicz Sobolev spaces in unbounded domains driven by a double phase operator with variable exponents that depend on the unknown solution and its gradient. Using these embeddings and an abstract critical point theorem, we prove the existence and multiplicity of weak solutions for such problems associated with this new operator in the whole space $\mathbb{R}^d$. This work can be seen as a continuation of the recent paper by Bahrouni--Bahrouni--Missaoui--R\u{a}dulescu \cite{Bahrouni-Bahrouni-Missaoui-Radulescu-2024}.