ABSTRACT

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularity $\begin{array}{rl}(-\mathrm{\Delta }{)}_{p(\cdot )}^{s}u& =\frac{\lambda }{|u{|}^{\gamma \left(x\right)-1}u}+f(x,u)\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =0\mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N,N\ge 2$ is a smooth, bounded domain, $\lambda >0$, $s\in (0,1)$, $\gamma \left(x\right)\in (0,1)$ for all $x\in \overline{\mathrm{\Omega }}$, $N>sp(x,y)$ for all $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$ and $(-\mathrm{\Delta }{)}_{p(\cdot )}^s$ is the fractional $p(\cdot )$-Laplacian operator with variable exponent. The nonlinear function f satisfies certain growth conditions. Moreover, we establish a uniform $L^{\mathrm{\infty }}\left(\overline{\mathrm{\Omega }}\right)$ estimate of the solution(s) by the Moser iteration technique.

中文翻译：

---

具有奇异非线性的变指数椭圆偏微分方程的无穷多个小解

摘要

我们证明以下涉及奇点的非局部椭圆偏微分方程存在无穷多个非负解 $\begin{array}{rl}(-\mathrm{\Delta }{)}_{磷(\cdot )}^{秒}你& =\frac{\lambda }{|你{|}^{\gamma \left(X\right)-1}你}+F(X,你)\mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ 你& =0\mathrm{一世}\mathrm{n}{\mathbb{电阻}}^N\setminus \mathrm{\Omega },\end{array}$ 在哪里 $\mathrm{\Omega }\subset {\mathbb{电阻}}^N,N\ge 2$ 是一个光滑的有界域， $\lambda >0$, $秒\in (0,1)$, $\gamma \left(X\right)\in (0,1)$ 对所有人 $X\in \overline{\mathrm{\Omega }}$, $N>秒磷(X,是)$ 对所有人 $(X,是)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$ 和 $(-\mathrm{\Delta }{)}_{磷(\cdot )}^{秒}$ 是小数 $磷(\cdot )$-具有可变指数的拉普拉斯算子。非线性函数f满足一定的生长条件。此外，我们建立了统一的${升}^{\mathrm{\infty }}\left(\overline{\mathrm{\Omega }}\right)$ 通过 Moser 迭代技术估计解。