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

Abstract

We are concerned with the following nonlinear Schrödinger equation $$\begin{aligned} -\varepsilon ^2\Delta u+ V(x)u=|u|^{p-2}u,\quad u\in H^1(\mathbb {R}^N), \end{aligned}$$ where $$N\ge 3$$ , $$2<p<\frac{2N}{N-2}$$ . 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. Especially, the degeneracy of critical points is allowed in this paper.