Normalized solutions of mass supercritical Schr\"odinger equations with potential

TL;DR

This paper establishes the existence of normalized solutions for a class of mass supercritical nonlinear Schrödinger equations with various potentials, using a novel min-max approach.

Contribution

It introduces a new min-max method to prove the existence of normalized solutions for supercritical Schrödinger equations with diverse potentials.

Findings

Existence of solutions with prescribed mass under various potential conditions

Solutions exist for potentials with singularities and decay at infinity

New min-max technique developed for this class of problems

Abstract

This paper is concerned with the existence of normalized solutions of the nonlinear Schr\"odinger equation \[ -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We prove the existence of a solution $(u,\lambda)\in H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=\rho$ under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.