  The aim of this paper is to treat the following problem $$ (P) \left\{
\begin{array}{rcll} (-\Delta)^s_{p, \beta} u &= & f(x,u) &\mbox{ in }\Omega, u
& = & 0 &\mbox{ in } \mathds{R}^N\setminus\Omega, \end{array} \right. $$ where
  $$ (-\Delta)^s_{p,\beta}\, u(x):=P.V.
\int_{\mathds{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}
\frac{dy}{|x|^\beta|y|^\beta},$$ $\Omega$ 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 $f\gneq 0$, 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 $f\gneq 0$, then the solution $u$ satisfies a suitable weak
Harnack inequality.
