Quasilinear elliptic equations with sub-natural growth terms in bounded domains

Abstract

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.