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A Struwe-type Decomposition Result for Weighted Critical $p$-Laplace Equations

Edward Chernysh

Abstract

We establish Struwe-type decompositions of Palais-Smale sequences for a class of critical $p$-Laplace equations of the Caffarelli-Kohn-Nirenberg type in a bounded domain $Ω\subset\mathbb{R}^n$, $n\ge2$, containing the origin. In doing so, we highlight important differences introduced by the weights and require new rescaling laws to account for this new framework.

A Struwe-type Decomposition Result for Weighted Critical $p$-Laplace Equations

Abstract

We establish Struwe-type decompositions of Palais-Smale sequences for a class of critical -Laplace equations of the Caffarelli-Kohn-Nirenberg type in a bounded domain , , containing the origin. In doing so, we highlight important differences introduced by the weights and require new rescaling laws to account for this new framework.

Paper Structure

This paper contains 20 sections, 25 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Historical Context and Classifications
  3. Classifications
  4. Paper structure
  5. Homogeneous Sobolev Spaces and Palais-Smale Sequences
  6. Pointwise Convergence of the Gradients
  7. Palais-Smale Sequences and Energies of Critical Points
  8. Homogeneity and New Rescaling Transformations
  9. Modifying the Rescaling: A New Transform
  10. Transformations and the $L^q$-norm
  11. Locality
  12. Convergence of Domains
  13. Energy and the New Transforms
  14. Locality
  15. Iterative results and Solution Extraction
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain containing the origin and let $(u_\alpha)$ be a Palais-Smale sequence for eq:probOm. Assume $a \neq b$ and let $\gamma > 0$ be the homogeneity exponent from eq:exponent. Then, there exists a subsequence $(u_\beta)$ of $(u_\alpha)$ along with Furthermore, as $\beta \to \infty$, we obtain

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 42 more