Doubly nonlocal system with Hardy–Littlewood–Sobolev critical nonlinearity
Résumé
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