Normalized ground states for the mass supercritical Schr\"{o}dinger-Bopp-Podolsky system: existence, uniqueness, limit behavior, strong instability

TL;DR

This paper investigates the existence, uniqueness, asymptotic behavior, and instability of normalized ground states for a mass supercritical Schrödinger-Bopp-Podolsky system with nonlocal nonlinearity, using variational methods and stability analysis.

Contribution

It establishes the existence of normalized ground states via mountain-pass arguments on $L^2$-spheres and analyzes their limit behavior and stability properties in the supercritical regime.

Findings

Normalized ground states exist for small $L^2$-sphere radii.

As mass vanishes or grows, ground states approach solutions of the classical Schrödinger equation.

Strong instability of certain standing waves is demonstrated.

Abstract

This paper concerns the normalized ground states for the nonlinear Schr\"{o}dinger equation in the Bopp-Podolsky electrodynamics. This equation has a nonlocal nonlinearity and a mass supercritical power nonlinearity, both of which have deep impact on the geometry of the corresponding functional, and thus on the existence, limit behavior and stability of the normalized ground states. In the present study, the existence of critical points is obtained by a mountain-pass argument developed on the $L^2$-spheres. To be specific, we show that normalized ground states exist, provided that spherical radius of the $L^2$-spheres is sufficiently small. Then, by discussing the relation between the normalized ground states of the Schr\"{o}dinger-Bopp-Podolsky system and the classical Schr\"{o}dinger equation, we show a precise description of the asymptotic behavior of the normalized ground states as the mass vanishes or tends to infinity. Moreover, we discuss the radial symmetry and uniqueness of the normalized ground states. Finally, the strong instability of standing waves at the mountain-pass energy level is studied by constructing an equivalent minimizing problem. Also, as a byproduct, we prove that the mountain-pass energy level gives a threshold for global existence based on this equivalent minimizing problem.