Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries

TL;DR

This paper proves the existence of sign-changing solutions with multiple bubbles for a slightly subcritical elliptic problem in symmetric convex domains, highlighting the role of domain symmetry in solution structure.

Contribution

It establishes the existence of multi-bubble nodal solutions in symmetric convex domains for the subcritical elliptic problem, extending previous results to sign-changing solutions with multiple bubbles.

Findings

Existence of four-bubble nodal solutions with two positive and two negative bubbles.

Solutions exist in convex domains with certain symmetries.

Analysis of solutions as the parameter epsilon approaches zero.

Abstract

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-\Delta u=|u|^{2^*-2-\e}u \hbox{in}\Omega, \quad u=0 \hbox{on}\partial \Omega,$$ where $\Omega$ is a smooth bounded domain in $\R^N$, $N\geq 3$, $2^*=\frac{2N}{N-2}$ and $\e>0$ is a small parameter. In particular we prove that if $\Omega$ is convex and satisfies a certain symmetry, then a nodal four-bubble solution exists with two positive and two negative bubbles.