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

Abstract

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -Δu&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }Ω,\\ u&=0, &\hbox{ on }\partialΩ. \end{array}\right. \eqno(P) \end{equation*} on the annulus $\Omega :=\{x\in {\mathbb{R}}^N:a<|x|<b\}$, $N\ge 3.$ 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 $r_0^{\frac{N-2}{2}}{u}_0(r_0)=:\underset{a\le r\le b}{\max }{r}^{\frac{N-2}{2}}{u}_0(r).$