Multiple normalized solutions for quasi-linear Schr\"odinger equations

TL;DR

This paper establishes the existence of two distinct solutions with prescribed $L^2$-norm for a quasi-linear Schrödinger equation, using a perturbation method to handle the non-differentiability of the functional.

Contribution

It introduces a novel approach to find multiple solutions for quasi-linear Schrödinger equations with prescribed norm, overcoming non-differentiability issues.

Findings

Existence of two solutions with prescribed $L^2$-norm

One solution is a mountain pass solution, the other is a local or global minimum

Utilizes a perturbation method to address non-differentiability

Abstract

In this paper we prove the existence of two solutions having a prescribed $L^2$-norm for a quasi-linear Schr\"odinger equation. One of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global. To overcome the lack of differentiability of the associated functional, we rely on a perturbation method developed in [27].