A new type of bubble solutions for a Schrödinger equation with critical growth

Abstract

In this paper, we investigate the following critical elliptic equation $$-\Delta u+V(y)u=u^{\frac{N+2}{N-2}},u>0,\text{in}{\mathbb{R}}^N,u\in H^1({\mathbb{R}}^N),$$ where $V(y)$ is a bounded non-negative function in ${\mathbb{R}}^N.$ Assuming that $V(y)=V(|\hat{y}|,y^{\ast }),y=(\hat{y},y^{\ast })\in {\mathbb{R}}^4\times {\mathbb{R}}^{N-4}$ and gluing together bubbles with different concentration rates, we obtain new solutions provided that $N\ge 7,$ whose concentrating points are close to the point $(r_0,y_0^{\ast })$ which is a stable critical point of the function $r^{2}V(r,y^{\ast })$ satisfying $r_0>0$ and $V(r_0,y_0^{\ast })>0.$ In order to construct such new bubble solutions for the above problem, we first prove a non-degenerate result for the positive multi-bubbling solutions constructed in \cite{PWY-18-JFA} by some local Pohozaev identities, which is of great interest independently. Moreover, we give an example which satisfies the assumptions we impose.

36pages