Existence and stability of standing waves for nonlinear Schrodinger systems involving the fractional Laplacian

2016/04/06 by Santosh Bhattarai, Bhattarai, Santosh
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1604.01718

22 pages

arxiv created 2016/04/06 · arxiv updated 2016/04/07

Abstract

In the present paper we consider the coupled system of nonlinear Schrödinger equations with the fractional Laplacian \ \beginaligned (-Δ)^αu₁ = λ₁u₁+f₁(u₁)+∂₁F(u₁,u₂) in ℝ^N,
(-Δ)^αu₂ = λ₂u₂+f₂(u₂)+∂₂F(u₁,u₂) in ℝ^N, \endaligned . where u₁, u₂:ℝ^N→ ℂ, N≥ 2, and 0<α<1. By studying an appropriate family of constrained minimization problems, we obtain the existence of solutions in the space H^α(ℝ^N) × H^α(ℝ^N) satisfying ∫_ℝ^N|u₁|² dx = σ₁ \textrmand ∫_ℝ^N|u₂|² dx=σ₂ for given σ_j>0. The numbers λ₁ and λ₂ in the system appear as Lagrange multiplier. The method is based on the concentration compactness arguments, but introduces a new way to verify some of the properties of the variational problem that are required in order for the concentration compactness method to work. We consider the case when f_j(s)=μ_j|s|^p_j-2s and F(s,t)=β|s|^r₁|t|^r₂ with μ_j>0, β>0, and the values r_i>1, 2<p_j, r₁+r₂<2+(4α)/(N). The method also enables us to prove the stability result of standing wave solutions associated with the set of global minimizers.

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