In this paper, we study the existence and regularity of a non-trivial weak solution to the following semilinear elliptic problem with singular weights and singular nonlinearity: $\{\begin{array}{ll}{\displaystyle -div(|x{|}^{-2\beta }\mathrm{\nabla }w)-\mu \frac{w}{|x{|}^{2(\beta +1)}}=\frac{g(x)}{w^{{\theta }_1}}+\frac{f(x)}{w^{{\theta }_2}}}& \mathrm{in}\mathrm{\Omega },\\ {\displaystyle w>0}& \mathrm{in}\mathrm{\Omega },\\ {\displaystyle w=0}& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$ under assumptions ${\theta }_1\ge {\theta }_2>0,0<\mu <{\left(\frac{N-2(\beta +1)}{2}\right)}^2$ and $0\le g\in L^k(\mathrm{\Omega }),$ $0\le f\in L^k(\mathrm{\Omega }),1<k<\frac{N}{2},0<\beta <\left(\frac{N-2}{2}\right)$.

中文翻译：

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奇异半线性椭圆问题的正解

在本文中，我们研究了以下具有奇异权重和奇异非线性的半线性椭圆问题的非平凡弱解的存在性和规律性：$\{\begin{array}{ll}{\displaystyle -d\mathrm{一世}v(|X{|}^{-2\beta }\mathrm{\nabla }w)-\mu \frac{w}{|X{|}^{2(\beta +1)}}=\frac{G(X)}{w^{{\theta }_1}}+\frac{F(X)}{w^{{\theta }_2}}}& \mathrm{在}\mathrm{\Omega },\\ {\displaystyle w>0}& \mathrm{在}\mathrm{\Omega },\\ {\displaystyle w=0}& \mathrm{上}\mathrm{\partial }\mathrm{\Omega },\end{array}$在假设下${\theta }_1\ge {\theta }_2>0,0<\mu <{\left(\frac{ñ-2(\beta +1)}{2}\right)}^2$和$0\le G\in {\mathrm{大号}}^{ķ}(\mathrm{\Omega }),$ $0\le F\in {\mathrm{大号}}^{ķ}(\mathrm{\Omega }),1<ķ<\frac{ñ}{2},0<\beta <\left(\frac{ñ-2}{2}\right)$.