Normalized solutions for the NLS equation with potential in higher dimension: the purely Sobolev critical case

TL;DR

This paper investigates normalized solutions to the higher-dimensional NLS equation with potential and Sobolev critical nonlinearity, establishing existence results for mountain-pass solutions and local minimizers under new assumptions.

Contribution

It introduces new techniques and assumptions to find mountain-pass and local minimizer solutions for the NLS equation in higher dimensions, addressing open problems.

Findings

Existence of mountain-pass solutions for N≥6.

Existence of local minimizers with negative energy for N≥3.

Improved results over previous studies by Verzini and Yu.

Abstract

We study normalized solutions for the nonlinear Schrodinger (NLS) equation with potential and Sobolev critical nonlinearity. By establishing suitable assumptions on the potential, together with new techniques, we find a mountain-pass type solution for N>=6, which solves an open problem presented in a recent paper [Verzini and Yu, arXiv:2505.05357v1]. Moreover, we also find a local minimizer with negative energy for N>=3, which improves the results in [Verzini and Yu, arXiv:2505.05357v1].