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Global compactness results for fractional $p$-Laplace Hardy Sobolev operator on a bounded domain

Nirjan Biswas

TL;DR

This work analyzes global compactness for a nonlinear critical Hardy-Sobolev problem driven by the fractional $p$-Laplace operator on a bounded domain. Building on Struwe-type ideas, it proves a complete Palais–Smale decomposition of PS sequences into a weak limit solving the original equation and finitely many bubble profiles solving limiting equations on $\mathbb{R}^d$, with precise energy and norm splitting. A key novelty is the role of the Hardy potential and the parameter $\alpha$, which yields either one or two bubble types depending on $\alpha$, and leads to a clear separation of scales and centers ensuring bubble decoupling. The results extend known global compactness frameworks to the nonlocal fractional $p$-Laplacian with Hardy-Sobolev structure and provide a pathway to obtain positive weak solutions via constrained minimization on the Nehari manifold.

Abstract

In this paper, we establish a Struwe type global compactness result for a class of nonlinear critical Hardy-Sobolev exponent problems driven by the fractional $p$-Laplace Hardy-Sobolev operator.

Global compactness results for fractional $p$-Laplace Hardy Sobolev operator on a bounded domain

TL;DR

This work analyzes global compactness for a nonlinear critical Hardy-Sobolev problem driven by the fractional -Laplace operator on a bounded domain. Building on Struwe-type ideas, it proves a complete Palais–Smale decomposition of PS sequences into a weak limit solving the original equation and finitely many bubble profiles solving limiting equations on , with precise energy and norm splitting. A key novelty is the role of the Hardy potential and the parameter , which yields either one or two bubble types depending on , and leads to a clear separation of scales and centers ensuring bubble decoupling. The results extend known global compactness frameworks to the nonlocal fractional -Laplacian with Hardy-Sobolev structure and provide a pathway to obtain positive weak solutions via constrained minimization on the Nehari manifold.

Abstract

In this paper, we establish a Struwe type global compactness result for a class of nonlinear critical Hardy-Sobolev exponent problems driven by the fractional -Laplace Hardy-Sobolev operator.
Paper Structure (3 sections, 7 theorems, 113 equations)

This paper contains 3 sections, 7 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Global compactness results

Key Result

Theorem 1.1

Let $s\in (0,1), p \in (1, \infty), \mu \in (0,\mu_{d,s,p})$, and $\alpha \in [0,sp)$. Let $\Omega$ be a bounded open set containing origin and $a \in L^{\frac{d-\alpha}{sp-\alpha}}(\Omega)$. Let $\{u_n\}$ be a (PS) sequence for $I_{\mu,a,\alpha}$ at level $\eta$. Then there exists a subsequence (st and $\tilde{U}_j$ weakly satisfies such that where $o_n(1) \rightarrow 0$ as $n \rightarrow \inft

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 3 more