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