EXISTENCE OF SOLUTIONS FOR NON-LOCAL ELLIPTIC SYSTEMS WITH HARDY-LITTLEWOOD-SOBOLEV CRITICAL NONLINEARITIES

In this work, we establish the existence of solutions for the non- linear nonlocal system of equations involving the fractional Laplacian, (-Delta)(s)u = au + bv + 2p/p + q integral(Omega) vertical bar v(y)vertical bar(q)/vertical bar x - y vertical bar(mu) dy vertical bar u vertical bar(p-2)u +2 xi(1) integral(Omega) vertical bar u(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 = bu + cv + 2q/p + q integral(Omega) vertical bar u(y)vertical bar(p)/vertical bar x - y vertical bar(mu) dy vertical bar v vertical bar(q-2)v +2 xi(2) integral(Omega) vertical bar v(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 (-Delta)(s) is the fractional Laplacian operator, Omega is a smooth bounded domain in R-N, 0 < s < 1, N > 2s, 0 < mu < N, > 0, xi(1), xi(2) >= 0, 1 < p, q <= 2(mu)* and 2(mu)* = 2N-mu/N-2s is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality. The nonlinearities can interact with the spectrum of the fractional Laplacian. More specifically, the interval defined by the two eigenvalues of the real matrix from the linear part contains an eigenvalue of the spectrum of the fractional Laplacian. In this case, resonance phenomena can occur.