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

Abstract

The aim of this paper is to study the following problem: (equation presented) where ω is a smooth bounded domain of ℝ N containing the origin, with 0 ≤ β < p < N, s ϵ (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, σ) = 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 ϵ L 1 (ω). In addition, if f ≤ 0, we show that the problem above has a unique entropy positive solution. (ii) f(x, σ) = λσq + g(x), σ ≤ 0; in this case, according to the values of λ and q, we get the largest class of data g for which the problem above has a positive solution.