We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta _{p, w} u = \sigma u^{q} &{} \text {in }\Omega ,\\ u = 0 &{} \text {on }\partial \Omega \end{array}\right. } \end{aligned}$$

in the sub-natural growth case \(0< q < p - 1\), where \(\Omega \) is a bounded domain in \({\mathbb {R}}^{n}\), \(\Delta _{p, w}\) is a weighted p-Laplacian, and \(\sigma \) is a nonnegative (locally finite) Radon measure on \(\Omega \). We give criteria for the existence problem. For the proof, we investigate various properties of p-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.

中文翻译：

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有界域中具有次自然增长项的拟线性椭圆方程

我们考虑以下类型的加权拟线性椭圆微分方程的正解的存在

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta _{p, w} u = \sigma u^{q} &{} \text {in }\Omega ,\ \ u = 0 &{} \text {on }\partial \Omega \end{array}\right. } \end{对齐}$$

在次自然增长的情况下\(0< q < p - 1\)，其中\(\Omega \)是\({\mathbb {R}}^{n}\) 中的有界域，\(\ Delta _{p, w}\)是加权p -拉普拉斯算子，而\(\sigma \)是\(\Omega \)上的非负（局部有限）氡测量。我们给出存在问题的标准。为了证明，我们研究了p超谐波函数的各种性质，特别是具有无限测量数据的 Dirichlet 问题的可解性。