Normalized solutions on large smooth domains to the Schr\"{o}dinger equations with potential and combined nonlinearities: The Sobolev critical case

TL;DR

This paper investigates the existence and multiplicity of normalized solutions to nonlinear Schrödinger equations with potential and combined nonlinearities on large smooth domains, focusing on the Sobolev critical case.

Contribution

It extends the understanding of normalized solutions in the Sobolev critical case, addressing an open problem and overcoming limitations of the Pohozaev identity approach due to potential presence.

Findings

Established existence of solutions in the Sobolev critical case.

Provided multiplicity results for prescribed mass solutions.

Extended previous work to more general nonlinearities and domains.

Abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrodinger equations with mixed nonlinearities. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential. Besides, Our study can be regarded as a Sobolev critical case complement of Bartsch-Qi-Zou (Math Ann 390, 4813-4859, 2024), which has addressed an open problem raised in Bartsch et al.(Commun Partial Differ Equ 46(9):1729--1756,2021).