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

Published 5 Apr 2018 in math.AP | (1804.01687v1)

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 $r_0^{{\frac{N-2}{2}}u_0(r_0)=:\displaystyle\max_{a\leq} r\leq b}r^{{\frac{N-2}{2}}u_0(r).$}