Normalized multi-bump solutions of nonlinear Schrödinger equations via variational approach

Abstract

We investigate the existence and concentration behavior of the multi-bump solutions for the nonlinear Schrödinger equation $-\hbar ^2\Delta v-K(x)|v|^{2\sigma }v=-\lambda v$ with an $L^2$-constraint in the $L^2$-subcritical case $\sigma \in (0,\frac{2}{N})$ and the $L^2$-supercritical case $\sigma \in (\frac{2}{N},\frac{2^*}{N})$, where $N\ge 1$ is the dimension, $\hbar >0$ is a small parameter and $K>0$ possesses several local maximum points. By variational approach, we construct normalized multi-bump solutions concentrating at a finite set of local maximum points of K. The construction combines the variational gluing arguments of Séré (Math Z 209(1):27–42, 1992) and Coti-Zelati and Rabinowitz (J Am Math Soc 4(4):693–727, 1991, Commun Pure Appl Math 45(10):1217–1269, 1992) and a penalization technique which is developed in order not to solve local minimization problems on the $L^2$ sphere. Our approach is robust without imposing any nondegeneracy assumptions on K.