Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations

Abstract

We study the following nonlinear Schrödinger equation $$i{u}_t=-\Delta u+V(x)u-a|u{|}^{q}u(t,x)\in {\mathbb{R}}^1\times {\mathbb{R}}^2,$$ where $a>0,q\in (0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^{\ast }:=\Vert Q{\Vert }_2^2$, where $Q$ is the unique (up to translations) positive radial solution of $\Delta u-u+u^3=0$ in ${\mathbb{R}}^2$, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of the standing waves for the above equation as $q\nearrow 2$.

18 pages

Cited by in corpus (5)

• Concentration Behavior of Nonlinear Hartree-type Equation with almost Mass Critical Exponent
• Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential
• Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations
• On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials
• Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation