Normalized solutions to the mixed dispersion nonlinear Schr\"odinger equation in the mass critical and supercritical regime

TL;DR

This paper investigates the existence, multiplicity, and stability of solutions to a mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regimes, focusing on ground states and radial solutions.

Contribution

It provides new results on the existence and multiplicity of solutions, including ground states and positive solutions, under mass constraints in the critical and supercritical regimes.

Findings

Existence of ground states under certain conditions.

Multiplicity of radial solutions.

Stability analysis of standing waves.

Abstract

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\"odinger equation $$ \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $\gamma >0, N \geq 1, 4 \leq \sigma N < \frac{4N}{(N-4)^+}$, whereas the parameter $\alpha \in \R$ will appear as a Lagrange multiplier. Given $c \in \R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.