On the Fractional p-laplacian equations with weight and general datum

Abstract

The aim of this paper is to treat the following problem $$ (P) \left\{ \begin{array}{rcll} (-Δ)^s_{p, β} u &= & f(x,u) &\mbox{ in }Ω, u & = & 0 &\mbox{ in } \mathds{R}^N\setminusΩ, \end{array} \right. $$ where $$ (-Δ)^s_{p,β}\, u(x):=P.V. \int_{\mathds{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}} \frac{dy}{|x|^β|y|^β},$$ $\Omega$ is a bounded domain containing the origin, $0\le \beta <\frac{N-ps}{2}$, $1<p<N$, $s\in (0,1)$ with $ps<N$. The main result of this paper is to prove the existence of a weak solution under additional hypotheses on $f$. In particular, we will consider two cases: 1- $f(x,s)=f(x)$, in this case we prove the existence of a weak solution, that is in a suitable weighted fractional Sobolev spaces, for all $f\in L^1(\Omega )$. In addition, if $f⪈0$, we show that problem $(P)$ has a unique entropy positive solution. 2-$f(x,s)=\lambda s^q+g(x)$ , in this case, according to the values of $\lambda$ and $q$, we get the largest class of data $g$ for which problem $(P)$ has a positive solution. In the case where $f⪈0$, then the solution $u$ satisfies a suitable weak Harnack inequality.