On the fractional p-Laplacian equations with weight and general datum

The aim of this paper is to study the following problem: {Delta)(s)(p,beta)u = f(x, u) in Omega, u = 0 in R-N \ Omega, where Omega is a smooth bounded domain of R-N containing the origin, (-Delta)(s)(p,beta)u(x) := PV (RN)integral vertical bar u(x) - u(y)vertical bar(p-2) (u(x) - u(y))/vertical bar x - y vertical bar(N+ps) dy/vertical bar x vertical bar(beta)vertical bar y vertical bar(beta) with 0 <= beta < N-ps/2, 1 < p < N, s is an element of (0, 1), and ps < N. The main purpose of this work is to prove the existence of a weak solution under some hypotheses on f. In particular, we will consider two cases: (i) f(x, sigma) = f(x); in this case we prove the existence of a weak solution, that is, in a suitable weighted fractional Sobolev space for all f is an element of L-1(Omega). In addition, if f >= 0, we show that the problem above has a unique entropy positive solution. (ii) f(x, sigma) = lambda sigma(q) + g(x), sigma >= 0; in this case, according to the values of lambda and q, we get the largest class of data g for which the problem above has a positive solution.