Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation

Abstract

In any dimension $N\ge 1$, 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 Schrödinger equation in ${\mathbb{R}}^3$.

This version is the final one, corresponding to the paper now published in Math. Models Methods Appl. Sci. DOI: 10.1142/S0218202522500361

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