Normalized ground states for the NLS equation with combined nonlinearities

2020/01/01 by Nicola Soave · 30 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Nonlinear Photonic Systems

paper · doi:10.1016/j.jde.2020.05.016

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities −Δu=λu+μ|u|q−2u+|u|p−2uin RN, N≥1, having prescribed mass ∫RN|u|2=a2. Under different assumptions on q0 and μ∈R we prove several existence and stability/instability results. In particular, we consider cases when [Formula presented] i.e. the two nonlinearities have different character with respect to the L2-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature. We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.

Citations

• Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on ℝ³ R 3
• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• Orbital stability of standing waves for some nonlinear Schr�dinger equations
• Nonlinear Schr�dinger equations and sharp interpolation estimates
• Existence of solitary waves in higher dimensions
• Nonlinear scalar field equations, I existence of a ground state
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• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case

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• Uniqueness and stability of normalized ground states for Hartree equation with a harmonic potential
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