On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

We study positive solutions to the fractional Lane-Emden system {(-Delta)(s)u = v(p) + mu in Omega (-Delta)(s)v = u(q) + v in Omega (S) u = v = 0 in Omega(c) = R-N\Omega, where Omega is a C-2 bounded domains in R-N, s is an element of(0, 1), N > 2s, p > 0, q > 0 and mu, nu are positive measures in Omega. We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of mu and nu. Furthermore, if p, q is an element of (1, N+s/N-s), 0 <= mu, nu is an element of L-r (Omega) for some r > N/2s, we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.