Solution manifolds for some semilinear Schrodinger equations

Abstract

In this paper, we study the following semilinear Schr\"odinger system $$ -\triangle u+u=(1+K_\alpha(\epsilon x))|u|^{p-2}u\ in \mathbb{R}^N, u\in H^1(\mathbb{R}^N) $$ where $3\leq p<2^*$ and $\epsilon>0$, $\alpha>0$ are small parameters. Under some conditions on $K_\alpha$ and the parameters $\alpha$ and $\epsilon$, we show that this equation exist solution manifolds.