Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions

In this article, we study the existence and multiplicity of non-negative solutions of the following p-fractional equation: integral-2 integral R-n|u(y)-u(x)|p-2(u(y)-u(x)) |x-y|n+pady =dh(x)|u|(q-1) u+b(x)|u|(r-1) u in Omega, where Omega is a bounded domain in with continuous boundary, p a parts per thousand yen 2, n > p alpha, alpha a (0, 1), 0 < q < p -1 < r < p (au) - 1 with p (au) = np(n - p alpha)(-1), lambda > 0 and h, b are sign-changing continuous functions. We show the existence and multiplicity of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists lambda (0) such that for lambda a (0, lambda (0)), it has at least two non-negative solutions.