Infinitely many non-radial solutions to a critical equation on annulus

TL;DR

This paper constructs infinitely many non-radial, sign-changing solutions to a critical elliptic PDE on an annulus, using bubble arrangements to demonstrate complex solution structures.

Contribution

It introduces a method to generate infinitely many non-radial solutions with multiple bubbles arranged on a polygonal pattern for the critical problem on annuli.

Findings

Existence of infinitely many non-radial solutions.

Solutions feature multiple bubbles arranged on regular polygons.

Solutions exhibit sign-changing behavior.

Abstract

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -\Delta u&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }\Omega,\\ u&=0, &\hbox{ on }\partial\Omega. \end{array}\right. \eqno(P) \end{equation*} on the annulus $\Omega:=\{x\in \mathbb{R}^N: a<|x|<b\}$, $N\geq 3.$ In particular, for any integer $k$ large enough, we build a non-radial solution which look like the unique positive solution $u_0$ to $(P)$ crowned by $k$ negative bubbles arranged on a regular polygon with radius $r_0$ such that $r_0^{\frac{N-2}{2}}u_0(r_0)=:\displaystyle\max_{a\leq r\leq b}r^{\frac{N-2}{2}}u_0(r).$