标题： 一个涉及分数$p$-拉普拉斯算子和不连续临界非线性的奇异椭圆问题 显示英文标题

标题： A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

摘要： 在本文中，我们证明了一个具有奇点和不连续临界非线性的非线性非局部椭圆问题解的存在性，其形式如下。 \begin{align} \begin{split}\label{main_prob} (-\Delta)_p^su&=\mu g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega u&>0,~\text{in}~\Omega, u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega, \end{split} \end{align} 其中$\Omega\subset\mathbb{R}^N$是一个具有 Lipschitz 边界的有界区域，$s\in (0,1)$，$2<p<\frac{N}{s}$，$\gamma\in (0,1)$，$\lambda,\mu>0$， $\alpha\geq 0$是实数，$H$是 Heaviside 函数，即 $H(a)=0$如果$a\leq 0$，$H(a)=1$如果$a>0$且$p_s^*=\frac{Np}{N-sp}$是分数临界 Sobolev 指数。 在函数$g$的适当假设下，我们证明了该问题解的存在性。 此外，我们证明当 $\alpha\rightarrow0^+$时，对于每个这样的 $\alpha$，方程 $\eqref{main_prob}$的解序列收敛到一个问题的解，其中 $\alpha=0$。 显示英文摘要

摘要： In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob} (-\Delta)_p^su&=\mu g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega u&>0,~\text{in}~\Omega, u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega, \end{split} \end{align} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2<p<\frac{N}{s}$, $\gamma\in (0,1)$, $\lambda,\mu>0$, $\alpha\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ if $a>0$ and $p_s^*=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent. Under suitable assumptions on the function $g$, we prove the existence of solution to the problem. Furthermore, we show that as $\alpha\rightarrow0^+$, the sequence of solutions of $\eqref{main_prob}$ for each such $\alpha$ converges to a solution of the problem for which $\alpha=0$.