论文标题
加权分数问题，涉及单数非线性和$ l^1 $数据
A weighted fractional problem involving a singular nonlinearity and a $L^1$ data
论文作者
论文摘要
在本文中，我们显示了以下问题的独特熵解决方案的存在：\ begin {equination} \ begin {split}（-Δ）_ {p，α}^su＆= f（x） 0〜 \ text {in}〜\ Mathbb {r}^n \setMinusΩ，\ nonumber \ end {split} \ end {equication}，其中域$ω\ subset \ subset \ subset \ subset \ subset \ subset \ mathbb {r^n $是界限的，并包含origin，$ al in oink，$α\ in [0. 0，\ frac} n.2 ps} $ s \ in（0,1）$，$ 2- \ frac {s} {n} <p <p <\ infty $，$ sp <n $，$ g \ in l^1（ω）$，$ f \ in l^q（ω）$ for $ q> 1 $ and $ h $ for $ h $ in live y in $ prive $ prive $ prive $ pract $ pract $ pract。 $$（ - δ）_ {p，α}^su（x）= \ text {p。 v。} \ int _ {\ mathbb {r}^n} \ frac {| u（x）-u（y）-u（y）|^{p-2}（u（x）-u（y（y））} {| x-y |^{n+ps}}}} \ Mathbb {r}^n。$$
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 $Ω\subset \mathbb{R}^N$ is bounded and contains the origin, $ α\in[0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N}<p<\infty$, $sp<N$, $g\in L^1(Ω)$, $f\in L^q(Ω)$ for $q>1$ and $h$ is a general singular function with singularity at 0. Further, the fractional $p$-Laplacian with weight $α$ is given by $$(-Δ)_{p,α}^su(x)=\text{P. V.}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\frac{dy}{|x|^α|y|^α},~\forall x\in \mathbb{R}^N.$$
