Solutions in Lebesgue spaces to nonlinear elliptic equations with sub-natural growth terms

Abstract

We study the existence problem for positive solutions $u\in L^r({\mathbb{R}}^n)$, $0<r<\infty$, to the quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q<p-1$, where ${\Delta }_{p}u=\text{div}(|\nabla u{|}^{p-2}\nabla u)$ is the $p$-Laplacian with $1<p<\infty$, and $\sigma$ is a nonnegative measurable function (or measure) on ${\mathbb{R}}^n$. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators such as the $\mathcal{A}$-Laplacian $\text{div}\mathcal{A}(x,\nabla u)$, and the fractional Laplacian $(-\Delta {)}^{\alpha }$ on ${\mathbb{R}}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\text{div}(\mathcal{A}\nabla u)$ on an arbitrary domain $\Omega \subseteq {\mathbb{R}}^n$ with a positive Green function.

20 pages