Abstract
This paper deals with the following singular system:
{(-Delta)(p)(s)u + (-Delta)(q)(s) u = lambda f(x) vertical bar u vertical bar(r-2)u + 1-alpha/2-alpha-beta h(x) vertical bar u vertical bar(-alpha) vertical bar v vertical bar(1-beta) in Omega,
(-Delta)(p)(s)v + (-Delta)(q)(s) v = mu g(x) vertical bar-x vertical bar(r-2)v + 1-beta/2-alpha-beta h(x) vertical bar u vertical bar(1-alpha) vertical bar v vertical bar(1-beta) in Omega,
u=v=0 in R-N \ Omega.
where Omega subset of R-N is a bounded smooth domain, lambda, mu are positive parameters, s is an element of(0, 1), 1 < p < N/s, 0 < alpha, beta < 1, 2 - alpha - beta < q < p < r < p(s)* = Np /(N - sp), and (-Delta)sigma(s)u denotes the fractional sigma-Laplacian, sigma =p, q. Under appropriate conditions on the weight functions f,g, h which may change sign in Omega, we establish the existence of multiple solutions by using the Nehari manifold method. Our paper is one of the first attempts to study the existence of solutions for fractional singular systems involving sign-changing weight functions.