Existence results for singular elliptic problem involving a fractional p-Laplacian

In this article, the problem to be studied is the following {(-Delta)(p)(s)u +/- beta vertical bar u vertical bar(p-2)u/vertical bar x vertical bar(sp) = lambda f (x, u) in Omega (P +/-) u = 0 on R-N \Omega, Omega is a bounded regular domain in R-N (N >= 2) containing the origin, p > 1, s is an element of (0, 1), (N > ps), lambda > 0, 0 < beta < 1/c(H) where c(H) is the best constant in the fractional Hardy inequality, f : Omega x R -> R is a Caratheodory function satisfying a suitable growth condition and (-Delta)(p)(s) is the fractional p-Laplacian defined as (-Delta)(p)(s)u(x) := 2 lim epsilon -> 0 integral(RN\B epsilon(x)) vertical bar u(x) - u(y)vertical bar(p-2)(u(x) - u(y))/ vertical bar x - y vertical bar(N+sp) |x - y| N+sp dy, x. R-N , where B-epsilon(x) is the open e-ball of centre x and radius (epsilon). Using the critical point theory combining with the fractional Hardy inequality, we show that the problem (P+) admits at least two distinct nontrivial weak solutions. For the problem (P-), we use the concentration-compactness principle for fractional Sobolev spaces to give a weak lower semicontinuity result and prove that problem (P-) admits at least one non-trivial weak solution.