Existence and Multiplicity of Normalized Solutions to Schr\"{o}dinger Equations with General Nonlinearities in Bounded Domains

TL;DR

This paper establishes the existence of multiple normalized solutions, including ground states and nonradial solutions, for Schrödinger equations with general nonlinearities in bounded domains using variational methods.

Contribution

It introduces new variational techniques to prove the existence and multiplicity of normalized solutions, including nonradial solutions, in bounded domains.

Findings

Existence of two positive normalized solutions, including a ground state.

Proof of multiple solutions using Linking theorems in star-shaped domains.

Existence of nonradial normalized solutions in a ball.

Abstract

This paper focuses on the existence of multiple normalized solutions to Schr\"{o}dinger equations with general nonlinearities in bounded domains via variational methods. We first obtain two positive normalized solutions, one is a normalized ground state by searching for a local minimizer, and the other one is a mountain pass solution. Secondly, using a version of Linking theorems for normalized solutions, we prove the multiplicity of solutions to Schr\"{o}dinger equations in a star-shaped bounded domain. Moreover, we arrive at the existence of nonradial normalized solutions to Schr\"odinger equations in a ball.