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Quantization for sequences of blow-up solutions to an elliptic equation having nonlocal exponential nonlinearity

Mathew Gluck

Abstract

This work provides a description of the asymptotic behavior of sequences of solutions to an elliptic equation with a nonlocal exponential nonlinearity of Choquard type. The equation under consideration is a nonlocal analog of the classical prescribed Gaussian curvature equation. A concentration-compactness alternative is established for sequences of solutions to the equation under consideration whenever suitable integrability assumptions on the solutions and the curvature functions are satisfied. Under further regularity assumptions on the curvature functions, and when blow-up occurs in the concentration-compactness alternative, an energy quantization result is established.

Quantization for sequences of blow-up solutions to an elliptic equation having nonlocal exponential nonlinearity

Abstract

This work provides a description of the asymptotic behavior of sequences of solutions to an elliptic equation with a nonlocal exponential nonlinearity of Choquard type. The equation under consideration is a nonlocal analog of the classical prescribed Gaussian curvature equation. A concentration-compactness alternative is established for sequences of solutions to the equation under consideration whenever suitable integrability assumptions on the solutions and the curvature functions are satisfied. Under further regularity assumptions on the curvature functions, and when blow-up occurs in the concentration-compactness alternative, an energy quantization result is established.
Paper Structure (6 sections, 24 theorems, 151 equations)

This paper contains 6 sections, 24 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regularity for a single solution
  4. Proof of the Concentration-Compactness Alternative
  5. Proof of the $\sup +\inf$ inequality
  6. Quantization

Key Result

Theorem A

Let $\Omega\subset \mathbb R^2$ be a bounded domain and let $p\in (1, \infty]$. If $(V_k)_{k = 1}^\infty$ and $(u_k)_{k = 1}^\infty$ be sequences of functions on $\Omega$ for which $V_k\geq 0$ for all $k$, for which there exists a constant $C_0>0$ such that and for which is satisfied for all $k$, then there is a subsequence $(u_{k_\ell})_{\ell = 1}^\infty\subset (u_k)_{k = 1}^\infty$ for which o

Theorems & Definitions (44)

  • Theorem A
  • Theorem B
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem C
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • ...and 34 more