On Concentration Phenomena of Least Energy Solutions to Nonlinear Schrödinger Equations with Totally Degenerate Potentials

Abstract

We study concentration phenomena of the least energy solutions of the following nonlinear Schrödinger equation: \[ h^2 \Delta u - V(x) u + f( u ) = 0 \quad \text{in} \ \mathbb{R}^N, \ u>0, \ u \in H^1(\mathbb{R}^N)\,, \] for a totally degenerate potential $V$. Here $h>0$ is a small parameter, and $f$ is an appropriate, superlinear and Sobolev subcritical nonlinearity. In~[16], Lu and Wei proved that when the parameter $h$ approaches zero, the least energy solutions concentrate at the most centered point of the totally degenerate set $\Omega = \{ x \in \mathbb{R}^N \mid V(x) = \min_{ y \in \mathbb{R}^N } V(y) \}$ when $f(u) = u^p$. In this paper, we show that this kind of result holds for more general $f$. In particular, our proof does not need a so-called uniqueness-nondegeneracy assumption (see, the next-to-last paragraph in Section~1) on the limiting equation (2.6) in Section~2. Furthermore, in~[16] Lu and Wei made a technical assumption for $V$, that is, \[ V(x) - \min_{y \in \mathbb{R}^N} V(y) \geq C d(x, \partial \Omega)^2 \quad \text{for} \ x \in \Omega^{\rm c}\,, \] where $C$ is a positive constant, but our proof does not need this assumption. In our proof, we employ a modification of the argument which has been developed by del Pino and Felmer in~[9] using Schwarz's symmetrization.