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

The aim of this paper is to treat the following problem $$(P)\{\begin{array}{rclllll}(-\mathrm{\Delta }{)}_{p,\beta }^{s}u& =& f(x,u)& in\mathrm{\Omega },u& =& 0& in\text{mathds}{R}^N\setminus \mathrm{\Omega },\end{array}$$ where $$(-\mathrm{\Delta }{)}_{p,\beta }^{s}u(x):=P\mathrm{.}V\mathrm{.}{\int }_{\text{mathds}{R}^N}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{p-2}(u(x)-u(y))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{N+ps}}\frac{dy}{\mathrm{\mid }x{\mathrm{\mid }}^{\beta }\mathrm{\mid }y{\mathrm{\mid }}^{\beta }},$$ $\mathrm{\Omega }$ 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 $f\in L^1(\Omega)$\mathrm{.}Inaddition,if$f\gneq 0$,weshowthatproblem$(P)$ 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.