Abstract
We consider the nonlinear fractional problem (-Δ)su+V(x)u=f(x,u)inRN\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (-\\Delta )^{s} u + V(x) u = f(x,u)&\\quad \\hbox {in } \\mathbb {R}^N \\end{aligned}$$\\end{document}We show that ground state solutions converge (along a subsequence) in L^2_{mathrm {loc}} (mathbb {R}^N), under suitable conditions on f and V, to a ground state solution of the local problem as s rightarrow 1^-.
Highlights
The aim of this paper is to analyse the asymptotic behavior of least-energy solutions to the fractional Schrodinger problem:(−Δ)su + V (x)u = f (x, u) u ∈ Hs(RN ), in RN (1.1)under suitable assumptions on the scalar potential V : RN → R and on the nonlinearity f : RN ×R → R
We show that ground state solutions converge in L2loc(RN ), under suitable conditions on f and V, to a ground state solution of the local problem as s → 1−
We recall that the fractional laplacian is defined as the principal value of a singular integral via the formula: (−Δ)su(x) = C(N, s) lim ε→0 u(x) − u(y) RN \Bε(x) |x − y|N+2s dy with
Summary
The aim of this paper is to analyse the asymptotic behavior of least-energy solutions to the fractional Schrodinger problem:. Under suitable assumptions on the scalar potential V : RN → R and on the nonlinearity f : RN ×R → R. We recall that the fractional laplacian is defined as the principal value of a singular integral via the formula:. 1 − cos ζ1 |ζ |N +2s dζ1 · · · dζN. This formal definition needs a function space in which problem (1.1) becomes meaningful: we will come to this issue in Sect. 2. Several models have appeared in recent years that involve the use of the fractional laplacian.
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