On a fractional system of NLS-KDV equations with Hardy potentials

2023/09/18 by Rohit Kumar, Kumar, Rohit, Tuhina Mukherjee +3 · 1 citation
Mathematics · Physics and Astronomy · Social Sciences · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Race, History, and American Society

paper · pdf · doi:10.48550/arxiv.2309.09536

Abstract

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schrödinger-Korteweg-De Vries (NLS-KdV, in short) equations with Hardy potentials: \ \beginaligned (-Δ)^s₁ u - λ₁ \fracu|x|^2s₁ - u^2_s₁^*-1 amp;= 2νh(x) uv amp; in ~ ℝ^N, (-Δ)^s₂ v - λ₂ \fracv|x|^2s₂ - v^2_s₂^*-1 amp;= νh(x) u² amp; in ~ ℝ^N, u,v gt;0 in ~ ℝ^N ∖ \0\, \endaligned . where s₁,s₂ ∈ (0,1)~and~λ_i∈ (0, Λ_N,s_i) with Λ_N,s_i = 2 π^N/2 \fracΓ²((N+2s_i)/(4)) Γ((N+2s_i)/(2))Γ²((N-2s_i)/(4)) ~|Γ(-s_i)|, (i=1,2). By imposing certain assumptions on the parameter ν and on the function h, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.

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