The existence of a nontrivial solution in some ‘weaker’ sense of the system of equations: $\begin{array}{rl}{L}_{s}u+l\left(x\right)\varphi u+w\left(x\right)u^k& =\mu \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ (-\mathrm{\Delta }{)}^s\varphi & =l\left(x\right)u^2\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& >0\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =\varphi =0\mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega }\end{array}$ is proved. Here, $L_{s}u=(-\mathrm{\Delta }{)}^{s}u-\lambda u^{-\text{β}}$, $\lambda >0$, $0<\text{β}\le s<1$, l, w are bounded, nonnegative real-valued functions in Ω, μ is a nonnegative Radon measure and k>1 belongs to a certain range.

中文翻译：

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包含奇异非局部算子、奇异点和氡测度的系统的解的存在性

在方程组的某种“较弱”意义上的非平凡解的存在：$\begin{array}{rl}{\mathrm{大号}}_s你+l\left(X\right)\phi 你+w\left(X\right){你}^{ķ}& =\mu \mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ (-\mathrm{\Delta }{)}^s\phi & =l\left(X\right){你}^2\mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ 你& >0\mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ 你& =\phi =0\mathrm{一世}\mathrm{n}{\mathbb{R}}^{ñ}\setminus \mathrm{\Omega }\end{array}$被证明。这里，${\mathrm{大号}}_s你=(-\mathrm{\Delta }{)}^s你-\lambda {你}^{-\text{β}}$,$\lambda >0$,$0<\text{β}\le s<1$, l , w是有界的非负实值函数，单位为 Ω，μ是非负氡测量，k >1 属于某个范围。