Multiplicity Results of a Nonlocal Problem Involving Concave-Convex Nonlinearities

In this work, we investigate the following fractional p-Laplacian equation involving a concave-convex nonlinearities as follows, (P lambda){(-Delta)(p)(s)u = lambda u(q) + u(r) in Omega, u > 0 in Omega, u - 0 in RN\Omega, where Omega subset of R-N, N =>= 2 is a bounded domain with C(1,)1 boundary partial derivative Omega, lambda > 0, 1 < p < infinity, s is an element of (0, 1) such that N >= sp, 0 < q < p - 1 < r <= p(s)* - 1, p* s = Np/N-sp is the fractional critical Sobolev exponent and the nonlinear nonlocal operator (-Delta)(p)(s)u with s is an element of (0, 1) is the p-fractional Laplacian defined