Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

Abstract

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation $$\gamma {\Delta }^{2}u-\Delta u+\alpha u=|u{|}^{2\sigma }u,u\in H^2({\mathbb{R}}^N),$$ under the constraint $${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=c>0.$$ We assume $\gamma >0,N\ge 1,4\le \sigma N<\frac{4N}{(N-4{)}^{+}}$, whereas the parameter $\alpha \in \mathbb{R}$ will appear as a Lagrange multiplier. Given $c\in {\mathbb{R}}^{+}$, 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.

arXiv commentThe present version is accepted for publication in Trans. Amer. Math. Soc

References in corpus (7)

• Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations
• On dipolar quantum gases in the unstable regime
• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
• Dual variational methods and nonvanishing for the nonlinear Helmholtz equation
• Oscillating solutions for nonlinear Helmholtz Equations
• Singular solutions of the L^2-supercritical biharmonic Nonlinear Schrodinger equation
• Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation

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