Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms

TL;DR

This paper characterizes when finite energy solutions exist for certain quasilinear elliptic equations with sub-natural growth terms, providing explicit conditions, uniqueness, and sharp bounds using potential estimates.

Contribution

It offers necessary and sufficient conditions for solutions in Sobolev spaces for equations with sub-natural growth, extending to general quasilinear operators and employing potential theory tools.

Findings

Explicit condition on $\sigma$ for solution existence

Proof of solution uniqueness in the Sobolev space

Sharp bounds derived using Wolff potential estimates

Abstract

We study finite energy solutions to quasilinear elliptic equations of the type $$ -\Delta_pu=\sigma \, u^q \quad \text{in } \mathbb{R}^n,$$ where $\Delta_p$ is the $p$-Laplacian, $p>1$, and $\sigma$ is a nonnegative function (or measure) on $\mathbb{R}^n$, in the case $0<q < p-1$ ( below the "natural growth" rate $q=p-1$ ). We give an explicit necessary and sufficient condition on $\sigma$ which ensures that there exists a solution $u$ in the homogeneous Sobolev space $L_0^{1,p}(\mathbb{R}^n)$, and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the $\mathcal{A}$-Laplacian $ \text{div} \mathcal{A}(x,\nabla \cdot)$ in place of $\Delta_p$ are considered as well.