Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations

TL;DR

This paper establishes necessary and sufficient conditions for positive solutions of certain sublinear elliptic equations to satisfy pointwise estimates, extending classical results to more general operators and fractional Laplacians.

Contribution

It provides new criteria for the existence of solutions with Brezis-Kamin type estimates for a broad class of quasilinear and fractional elliptic equations.

Findings

Derived conditions for solution existence based on Wolff potentials.

Extended Brezis-Kamin estimates to fractional Laplacians.

Provided explicit estimates for radially symmetric measures.

Abstract

We study quasilinear elliptic equations of the type $$-\Delta_pu=\sigma \, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $\Delta_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $\sigma \ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian $(-\Delta )^{\alpha}$ for $0<\alpha<\frac{n}{2}$ are treated as well, including explicit estimates for radially symmetric $\sigma$. Our results are new even in the classical case $p=2$ and $\alpha=1$.