Pointwise estimates of solutions to nonlinear equations for nonlocal operators

Alexander Grigor'yan, Igor Verbitsky

Published 2017-07-30Version 1

We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type \begin{equation*} u(x) - \int_\Omega G(x, y) \, g(u(y)) d \sigma (y) = h, \end{equation*} where $(\Omega, \sigma)$ is a locally compact measure space, $G(x, y)\colon \Omega\times \Omega \to [0, +\infty]$ is a kernel, $h \ge 0$ is a measurable function, and $g\colon [0, \infty)\to [0, \infty)$ is a monotone function. This problem is motivated by the semilinear fractional Laplace equation \begin{equation*} (-\Delta)^{\frac{\alpha}{2}} u - g(u) \sigma = \mu \quad \text{in} \, \, \Omega, \quad u=0 \, \, \, \text{in} \, \, \Omega^c, \end{equation*} with measure coefficients $\sigma$, $\mu$, where $g(u)=u^q$, $q \in \mathbb{R} \setminus\{0\}$, and $0<\alpha<n$, in domains $\Omega \subseteq\mathbb{R}^n$, or Riemannian manifolds, with positive Green's function $G$.