A weighted fractional problem involving a singular nonlinearity and a $L^1$ data

Abstract

In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-Δ)_{p,α}^su&= f(x)h(u)+g(x) ~\text{in}~Ω,\\ u&>0~\text{in}~Ω,\\ u&= 0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{split} \end{equation} where the domain $\Omega \subset {\mathbb{R}}^N$ is bounded and contains the origin, $\alpha \in [0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N}<p<\infty$, $sp<N$, $g\in L^1(\Omega )$, $f\in L^q(\Omega )$ for $q>1$ and $h$ is a general singular function with singularity at 0. Further, the fractional $p$-Laplacian with weight $\alpha$ is given by $$(-\Delta {)}_{p,\alpha }^{s}u(x)=\text{P. V.}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))}{|x-y{|}^{N+ps}}\frac{dy}{|x{|}^{\alpha }|y{|}^{\alpha }},\forall x\in {\mathbb{R}}^N.$$

References in corpus (1)

• Finite and Infinite energy solutions of singular elliptic problems: Existence and Uniqueness