A new type of bubble solutions for a Schr\"{o}dinger equation with critical growth

TL;DR

This paper constructs new multi-bubble solutions for a critical elliptic Schrödinger equation with variable potential, by gluing bubbles with different concentration rates near stable critical points, extending previous results.

Contribution

It introduces a novel method to create multi-bubble solutions with varying concentration rates for a critical Schrödinger equation, using local Pohozaev identities.

Findings

Existence of new solutions with multiple bubbles near stable critical points.

Proof of non-degeneracy for positive multi-bubble solutions.

Provided an example satisfying the theoretical assumptions.

Abstract

In this paper, we investigate the following critical elliptic equation $$ -\Delta u+V(y)u=u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,\R^{N},\,\,u\in H^{1}(\R^{N}), $$ where $V(y)$ is a bounded non-negative function in $\R^{N}.$ Assuming that $V(y)=V(|\hat{y}|,y^{*}),y=(\hat{y},y^{*})\in \R^{4}\times \R^{N-4}$ and gluing together bubbles with different concentration rates, we obtain new solutions provided that $N\geq 7,$ whose concentrating points are close to the point $(r_{0},y^{*}_{0})$ which is a stable critical point of the function $r^{2}V(r,y^{*})$ satisfying $r_{0}>0$ and $V(r_{0},y^{*}_{0})>0.$ 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.