Nonlinear potential estimates for sublinear problems with applications to elliptic semilinear and quasilinear equations

TL;DR

This paper surveys recent nonlinear potential estimates for positive solutions of sublinear elliptic problems, providing existence criteria and applications to fractional Laplacian and p-Laplace equations.

Contribution

It introduces new bilateral pointwise estimates for solutions, leading to existence and uniqueness results for various nonlinear elliptic equations with measure data.

Findings

Bilateral pointwise estimates for solutions

Existence criteria for solutions in measure spaces

Applications to fractional and p-Laplace elliptic equations

Abstract

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega, \] where $0 < q < 1$, $\sigma\ge 0$ is a Radon measure in $\Omega$, $ f \ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $\Omega\times\Omega$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u \in L^q_{{\rm loc}} (\Omega, \sigma)$. Applications are considered to semilinear elliptic equations involving the (fractional) Laplacian, \[ (-\Delta)^{\frac{\alpha}{2}} u = \sigma u^q + \mu \quad \textrm{in} \,\, \Omega, \qquad u=0 \, \, \textrm{in} \,\, \Omega^c. \] Here $0<q<1$, $\mu, \sigma \ge 0$ are Radon measures, and $\Omega$ is a bounded uniform domain in ${\mathbb R}^n$, if $0 < \alpha \le 2$, or the entire space ${\mathbb R}^n$, a ball or half-space, if $0 < \alpha <n$. Analogues of these results are presented for elliptic equations involving the $p$-Laplace operator on the entire space ${\mathbb R}^n$, \[ -\Delta_p u = \sigma u^q + \mu \quad \textrm{in} \,\, {\mathbb R}^n, \qquad \liminf_{x\to \infty} u(x)=0, \] where $0<q<p-1$, and $\mu, \sigma \ge 0$ are Radon measures. More general quasilinear equations with $\mathcal{A}$-Laplace operators ${\rm div} \mathcal{A}(x, \nabla u)$ in place of $\Delta_p$ are covered as well.