Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold

TL;DR

This paper develops a novel analytical method to construct sign-changing solutions for nonlinear critical elliptic equations by gluing a bubble to a degenerate critical manifold, even without standard non-degeneracy conditions.

Contribution

It introduces a new analyticity-based approach for glueing solutions in degenerate settings, expanding the toolkit for elliptic PDE analysis.

Findings

Successfully constructed blowing-up sign-changing solutions

Extended the glueing method to degenerate critical manifolds

Provided new insights into solution structure for critical elliptic equations

Abstract

We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.