Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schr\"{o}dinger equation

TL;DR

This paper investigates positive energy solutions constrained by mass for a class of coercive energy functionals, providing new existence results and insights into the cubic-quintic nonlinear Schrödinger equation in three dimensions.

Contribution

It introduces a method to find constrained critical points at positive energy levels for coercive problems, including local minima and mountain pass solutions, applied to the cubic-quintic nonlinear Schrödinger equation.

Findings

Existence of constrained critical points at positive energy levels.

Construction of solutions as local minimizers or mountain pass levels.

Application to the cubic-quintic nonlinear Schrödinger equation in A3.

Abstract

In any dimension $N \geq 1$, for given mass $m > 0$ and when the $C^1$ energy functional \begin{equation*} I(u) := \frac{1}{2} \int_{\mathbb{R}^N} |\nabla u|^2 dx - \int_{\mathbb{R}^N} F(u) dx \end{equation*} is coercive on the mass constraint \begin{equation*} S_m := \left\{ u \in H^1(\mathbb{R}^N) ~|~ \|u\|^2_{L^2(\mathbb{R}^N)} = m \right\}, \end{equation*} we are interested in searching for constrained critical points at positive energy levels. Under general conditions on $F \in C^1(\mathbb{R}, \mathbb{R})$ and for suitable ranges of the mass, we manage to construct such critical points which appear as a local minimizer or correspond to a mountain pass or a symmetric mountain pass level. In particular, our results shed some light on the cubic-quintic nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$.