Normalized clustering peak solutions for Schr\"odinger equations with general nonlinearities

TL;DR

This paper constructs multi-peak solutions for a nonlinear Schrödinger equation with general nonlinearities, analyzing their clustering behavior near local maxima of the potential as the parameter tends to zero.

Contribution

It introduces a novel method for lower gradient estimates without relying on high regularity flows, addressing the challenge of non-uniqueness and nondegeneracy in the limiting system.

Findings

Peaks cluster near local maxima of V as ε→0

Existence of ground state solutions for the autonomous case

Ground state energy can be positive, with strict subadditivity due to concavity

Abstract

We are concerned with the normalized $\ell$-peak solutions to the nonlinear Schr\"{o}dinger equation \[ -\varepsilon^2\Delta v+V(x)v=f(v)+\lambda v,\quad \int_{\mathbb{R}^N}v^2 =\alpha \varepsilon^N. \] Here $\lambda \in \mathbb{R}$ will arise as a Lagrange multiplier, $V$ has a local maximum point, and $f$ is a general $L^2$-subcritical nonlinearity satisfying a nonlipschitzian property that $\lim_{s\to0} f(s)/s=-\infty$. The peaks of solutions that we construct cluster near a local maximum of $V$ as $\varepsilon\to0$. Since there is no information about the uniqueness or nondegeneracy for the limiting system, a delicate lower gradient estimate should be established when the local centers of mass of functions are away from the local maximum of $V$. We introduce a new method to obtain this estimate, which is significantly different from the ideas in del Pino and Felmer (Math. Ann. 2002), where a special gradient flow with high regularity is used, and in Byeon and Tanaka (J. Eur. Math. Soc. 2013 \& Mem. Amer. Math. Soc. 2014), where an extra translation flow is introduced. We also give the existence of ground state solutions for the autonomous problem, i.e., the case $V\equiv0$. The ground state energy is not always negative and the strict subadditive property of ground state energy here is achieved by strict concavity.