Multiple solutions of a nonlocal system with singular nonlinearities

In this work, we study the fractional Laplacian equation with singular nonlinearity: {(-Delta)(s)u = lambda a(x)vertical bar u vertical bar(q-2)u + 1-alpha/2-alpha-beta c(x)vertical bar u vertical bar(-alpha)vertical bar v vertical bar(1-beta) in Omega, (-Delta)(s)v = mu b(x)vertical bar v vertical bar(q-2)v + 1-beta/2-alpha-beta c(x)vertical bar u vertical bar(1-alpha)vertical bar v vertical bar(-beta) in Omega, u = v = 0 in R-N\Omega, where Omega is a bounded domain in R-n with smooth boundary partial differential partial derivative Omega, N > 2s, s is an element of (0, 1), 0 < alpha < 1, 0 < beta < 1, 1 < q < 2 < 2(s)*, 2(s)* = 2N/N-2s is the fractional Sobolev exponent, lambda,mu are two parameters, a,b,c is an element of C(<(Omega)over bar> ) are nonnegative weight functions, and (-Delta)(s) is the fractional Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter lambda and mu.