Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:467

Article

Giacomoni, J.¹  Mukherjee, T.²  Sreenadh, K.² 

[1] Univ Pau & Pays Adour, CNRS, E2S, LMAP UMR 5142, Ave Univ, F-64013 Pau, France

[2] Indian Inst Technol Delhi, Dept Math, New Delhi 110016, India

关键词: Nonlocal operator;    Fractional Laplacian;    Choquard equation;    Hardy-Littlewood-Sobolev critical exponent;   

DOI  :  10.1016/j.jmaa.2018.07.035

来源: Elsevier

【 摘 要 】

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality {(-Delta)(s)u = lambda vertical bar u vertical bar(q-2)u+ (integral(Omega)vertical bar v(y)vertical bar(2*mu)/vertical bar x-y vertical bar(mu)dy) vertical bar u vertical bar(2 mu*-2)u in Omega (-Delta)(s)v = delta vertical bar v vertical bar(q-2)v+ (integral(Omega)vertical bar u(y)vertical bar(2*mu)/vertical bar x-y vertical bar(mu)dy) vertical bar v vertical bar(2 mu*-2)v in Omega u = v = 0 in R-n \ Omega, where Omega is a smooth bounded domain in R-n, n> 2s, s is an element of(0,1), (-Delta)(S) is the well known fractional Laplacian, mu is an element of(0, n), 2(mu)*, = 2n -mu/n-2s is the upper critical exponent Hardy-Littlewood-Sobolev inequality, 1 < q < 2 and lambda, delta > 0 are real parameters. We study the fibering maps corresponding to the functional associated with (P-lambda.delta) and show that minimization over suitable subsets of Nehari manifold renders the existence of at least two non trivial solutions of (P-lambda,P-delta) for suitable range of lambda and delta. (C) 2018 Elsevier Inc. All rights reserved.

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