Tokyo Journal of Mathematics

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

Shun KODAMA

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.

Article information

Source
Tokyo J. Math., Volume 40, Number 2 (2017), 565-603.

Dates
First available in Project Euclid: 9 January 2018

https://projecteuclid.org/euclid.tjm/1515466839

Mathematical Reviews number (MathSciNet)
MR3743732

Zentralblatt MATH identifier
06855948

Citation

KODAMA, Shun. On Concentration Phenomena of Least Energy Solutions to Nonlinear Schrödinger Equations with Totally Degenerate Potentials. Tokyo J. Math. 40 (2017), no. 2, 565--603. https://projecteuclid.org/euclid.tjm/1515466839

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