New type of solutions for Schr\"odinger equations with critical growth

Abstract

We consider the following nonlinear Schr\"odinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial function in $C^1$, $N\ge 5$. By using a finite reduction argument, we show that if $r^2V(r)$ has either an isolated local maximum or an isolated minimum at $r_0>0$ with $V(r_0)>0$, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of $O(3)$.