In this note, we show that in some cases, via the use of Hardy-type inequality, there is a non-trivial nonnegative \(W^{1,p}(R^n)\) weak solution to quasi-linear elliptic problem with the p-Laplacian on \(R^n\) and with Hardy-type singularity term. We also study the behavior of solutions to the Poisson equation of p-Laplacian on the whole space and this Poisson equation has a close relationship with the Gelfand-type equation.

中文翻译：

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p-Laplacian的Poisson方程和非线性Hardy型问题

在本说明中，我们表明，在某些情况下，通过使用Hardy型不等式，对于拟线性椭圆形存在非平凡的非负\（W ^ {1，p}（R ^ n）\）弱解\（R ^ n \）上的p -Laplacian和Hardy型奇点的问题。我们还研究了p-Laplacian的Poisson方程在整个空间上解的行为，并且该Poisson方程与Gelfand型方程密切相关。