Blow up solutions for Sinh-Gordon equation with residual mass

TL;DR

This paper constructs blow-up solutions with residual mass for the Sinh-Gordon equation in bounded domains, revealing limitations of existing concentration-compactness theory for this class of problems.

Contribution

It is the first to demonstrate residual mass blow-up solutions for the Sinh-Gordon equation, highlighting new phenomena not covered by previous theories.

Findings

Constructed solutions with partial and asymmetric blow-up

Showed concentration-compactness theory cannot be extended to this problem

First example of residual mass phenomena in Sinh-Gordon equations

Abstract

We are concerned with the Sinh-Gordon equation in bounded domains. We construct blow up solutions with residual mass exhibiting either partial or asymmetric blow up, i.e. where both the positive and negative part of the solution blow up. This is the first result concerning residual mass for the Sinh-Gordon equation showing in particular that the concentration-compactness theory of Brezis-Merle can not be extended to this class of problems.