Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

N. Chems Eddine, M. A. Ragusa, D. D. Repovš

Abstract

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis-Nirenberg problem.

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Abstract

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis-Nirenberg problem.
Paper Structure (5 sections, 13 theorems, 114 equations)

This paper contains 5 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Functional framework
  3. An extension of the Lions concentration-compactness principle
  4. A class of nonlinear anisotropic elliptic equations with critical growth
  5. Proofs of Theorems \ref{['thm1']} and \ref{['thm2']}

Key Result

Theorem 1

Suppose that assumptions ($\textbf{A}_1$)- ($\textbf{A}_3$), ($\textbf{f}_1$)-($\textbf{f}_3$), and ($\textbf{H}$) hold. Then for all $\lambda\in\mathbb{R}$ and $\beta> 0$, problem e1.1 has at least one nontrivial weak solution.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Example 1
  • Example 2
  • Proposition 1: Kováčik and Rákosnı́k KR
  • Remark 1
  • Proposition 2: Diening et al. Dien, Edmunds and Rakosnik Edmu
  • Theorem 3
  • proof
  • Remark 2
  • ...and 13 more