A Lagrangian approach for prescribed mass solutions of cubic-quintic Schr\"odinger equations and $L^2$-supercritical problems

TL;DR

This paper introduces a Lagrangian framework to establish the existence of radially symmetric solutions for nonlinear scalar field equations with prescribed mass, including cubic-quintic and supercritical cases, without relying on traditional growth conditions.

Contribution

It develops a novel Lagrangian approach using a minimax function to find solutions, extending existence results for complex nonlinear Schrödinger equations beyond previous assumptions.

Findings

Proves existence of solutions related to local extrema of a minimax function.

Provides new existence results for cubic-quintic equations in 2D and 3D.

Improves upon prior results by removing global growth conditions.

Abstract

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb R^N$ ($N \geq 2$): $$ (*)_m \quad - \Delta u + \mu u = g(u) \quad \text{in}\ {\mathbb R}^N, \quad {1\over 2} \int_{{\mathbb R}^N} u^2\, dx = m,$$ where $g(s) \in C({\mathbb R},{\mathbb R})$, $m > 0$ and $\mu \in {\mathbb R}$ is an unknown Lagrangian multiplier. We take an approach using a Lagrangian formulation of $(*)_m$: $$J_m(\mu,u)={1\over 2}\int_{{\mathbb R}^N} |\nabla u|^2\,dx -\int_{{\mathbb R}^N} G(u)\,dx +\mu\left({1\over 2}\int_{{\mathbb R}^N} u^2\, dx-m\right) \in C^1((0,\infty)\times H_r^1({\mathbb R}^N), {\mathbb R})$$ and we give new general existence results through the function: $$ b_m:\, (0,\infty) \to {\mathbb R};\ \mu \mapsto \text{Mountain Pass minimax value for}\ (u\mapsto J_m(\mu,u)).$$ We will show the existence of solutions of $(*)_m$ related to local minima and local maxima of $b_m(\mu)$. As applications, we study cubic-quintic type equations and $L^2$-supercritical problems. In particular, when $N=2,3$, we show new existence results of normalized solutions without assuming global Ambrosetti-Rabinowitz type conditions, which partially improve the preceding results due to Jeanjean [24] and Jeanjean-Lu [26, 28].