Existence and concentration of solution for a non-local regional Schrödinger equation with competing potentials

2016/11/07 by Claudianor O. Alves, Alves, Claudianor O., César E. Torres Ledesma +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP #msc:35A15 #msc:35B38 #msc:35J60

paper · pdf · doi:10.48550/arxiv.1611.02056

arxiv created 2016/11/07 · arxiv updated 2016/11/08

Abstract

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation \ ε^2α(-Δ)_ρ^α u + Q(x)u = K(x)|u|^p-1u, in ℝⁿ,
u∈ H^α(ℝⁿ) . where ε is a positive parameter, 0< α< 1, 1<p<(n+2α)/(n-2α), n>2α; (-Δ)_ρ^α is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x)>0, ρ, Q, K are competing functions. We study the existence of ground state and we analyze the behavior of semi-classical solutions as ε→ 0.

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