Standing waves with prescribed $L^2$-norm to nonlinear Schr\"odinger equations with combined inhomogeneous nonlinearities

TL;DR

This paper studies standing wave solutions with prescribed $L^2$-norm for a nonlinear Schrödinger equation with combined inhomogeneous nonlinearities, establishing existence, stability, symmetry, and decay properties across different mass regimes.

Contribution

It provides new results on existence, stability, symmetry, and decay of solutions to inhomogeneous nonlinear Schrödinger equations under $L^2$ constraints, covering subcritical, critical, and supercritical cases.

Findings

Compactness of minimizing sequences in the subcritical case

Orbital stability of minimizers in the subcritical case

Existence and symmetry of solutions in critical and supercritical cases

Abstract

In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $\mu =\pm 1$, $2<q<p<{2(N-b)}/{(N-2)^+}$, $0<b<\min\{2, N\}$ and the parameter $\lambda \in \R$ appearing as Lagrange multiplier is unknown. In the mass subcritical case, we establish the compactness of any minimizing sequence to the minimization problem given by the underlying energy functional restricted on the constraint. As a consequence of the compactness of any minimizing sequence, orbital stability of minimizers is derived. In the mass critical and supercritical cases, we investigate the existence, radial symmetry and orbital instability of solutions. Meanwhile, we consider the existence, radial symmetry and algebraical decay of ground states to the corresponding zero mass equation with defocusing perturbation. In addition, dynamical behaviors of solutions to the Cauchy problem for the associated dispersive equation are discussed.