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

TL;DR

This paper analyzes the asymptotic behavior of solution sequences to a nonlocal elliptic equation with exponential nonlinearity, establishing concentration-compactness and energy quantization results under certain conditions.

Contribution

It introduces a nonlocal analog of the prescribed Gaussian curvature equation and provides a detailed asymptotic analysis including energy quantization for blow-up solutions.

Findings

Established a concentration-compactness alternative for solution sequences.

Proved energy quantization under regularity and blow-up conditions.

Characterized blow-up behavior for nonlocal exponential elliptic equations.

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.