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

Abstract

Abstract The aim of this paper is to study the following problem: \left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p,\beta}u&\displaystyle=f(x% ,u)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is a smooth bounded domain of {\mathbb{R}^{N}} containing the origin, (-\Delta)^{s}_{p,\beta}u(x):=\mathrm{PV}\int_{\mathbb{R}^{N}}\frac{\lvert u(x)% -u(y)\rvert^{p-2}(u(x)-u(y))}{\lvert x-y\rvert^{N+ps}}\frac{dy}{\lvert x\rvert% ^{\beta}\lvert y\rvert^{\beta}} with {0\leq\beta<\frac{N-ps}{2}} , {1<p<N} , {s\in(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\in L^{1}(\Omega)} . In addition, if {f\gneq 0} , we show that the problem above has a unique entropy positive solution. (ii) {f(x,\sigma)=\lambda\sigma^{q}+g(x)} , {\sigma\geq 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.