ABSTRACT

In this paper, we establish the existence of at least one positive solution to a singular elliptic equation with zero boundary data and critical Hardy–Sobolev exponent. We show that the existence of the positive solution in high dimension depends on the sign of the mean curvature of the boundary near zero and in the low dimensions, it depends on the mass of the domain.

中文翻译：

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具有Hardy–Sobolev指数的边界奇异半线性方程的正解

摘要

在本文中，我们建立了具有零边界数据和临界Hardy-Sobolev指数的奇异椭圆方程的至少一个正解的存在。我们表明，高维正解的存在取决于边界的平均曲率在零附近的符号，而在低维中，正解的存在取决于域的质量。