Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations

Daomin Cao, Shuanglong Li, Peng Luo

Published 2015-04-23Version 1

We are concerned with the following nonlinear Schr\"odinger equation \begin{equation*} -\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,~u\in H^1(\R^N), \end{equation*} where $N\geq 3$, $2<p<\frac{2N}{N-2}$. For $\varepsilon$ small enough and a class of $V(x)$, we show the uniqueness of positive multi-bump solutions concentrating at $k$ different critical points of $V(x)$ under certain assumptions on asymptotic behavior of $V(x)$ and its first derivatives near those points. The degeneracy of critical points is allowed in this paper.