The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications

Abstract

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu >0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_{\mu }$ allows us to derive its behavior in the two limits $\mu \to 0$ and $\mu \to {\mu }_{\ast }$ where ${\mu }_{\ast }$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $L^2$ mass of $u_{\mu }$ in terms of $\mu$, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $u_{\mu }$.

References in corpus (3)

• A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application
• Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{ä}cker model
• Ground States for a Stationary Mean-Field Model for a Nucleon

Cited by in corpus (17)

• Multiple normalized solutions for a Sobolev critical Schrödinger equation
• On global minimizers for a mass constrained problem
• Dilute Bose gas with three-body interaction: recent results and open questions
• Traveling waves for a nonlinear Schrödinger system with quadratic interaction
• Limit profiles for singularly perturbed Choquard equations with local repulsion
• Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential
• On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations
• On ground states for the 2D Schrodinger equation with combined nonlinearities and harmonic potential
• Non-existence and multiplicity of positive solutions for Choquard equations with critical combined nonlinearities
• On an intercritical log-modified nonlinear Schrodinger equation in two spatial dimensions
• Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities
• Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions
• On action ground states of defocusing nonlinear Schrödinger equations
• Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension
• On solitary wave solutions to dispersive equations with double power nonlinearities
• Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity
• Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation