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

Ala Eddine Bahrouni, Anouar Bahrouni, Patrick Winkert

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--Rădulescu \cite{Bahrouni-Bahrouni-Missaoui-Radulescu-2024}.

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

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 . This work can be seen as a continuation of the recent paper by Bahrouni--Bahrouni--Missaoui--Rădulescu \cite{Bahrouni-Bahrouni-Missaoui-Radulescu-2024}.

Paper Structure

This paper contains 11 sections, 26 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Musielak-Orlicz Sobolev spaces
  4. Functional setting
  5. Embedding results
  6. Continuous embedding of $W^{1,\mathcal{H}}(\mathbb{R}^d)$ (Theorem \ref{['Inj1']})
  7. Compact embedding of $W^{1,\mathcal{H}}_V({\mathbb R}^d)$
  8. Compact embedding of $W^{1,\mathcal{H}}_{\operatorname{rad}}({\mathbb R}^d)$ (Theorem \ref{['thms']})
  9. Application
  10. Some properties of the energy functional
  11. Concluding remarks, perspectives, and open problems

Key Result

Theorem 1.2

Let hypotheses H be satisfied. Then, the following hold: the embedding is continuous.

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1.2: Continuous embedding
  • Theorem 1.3: Compact embedding
  • Theorem 1.4: Compact embedding
  • Theorem 1.5: Compact embedding
  • Theorem 1.6: Strauss radial embedding
  • Theorem 1.7: Lions-type lemma
  • Example 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 47 more