Pointwise estimates of solutions to nonlinear equations for nonlocal operators

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 $(\mathrm{\Omega },\sigma )$ is a locally compact measure space, $G(x, y)\colon \Omega\times \Omega \to [0, +\infty]$isakernel,$h \ge 0$ is a measurable function, and $g\text{\hspace{0.17em}⁣}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ 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$0<\alpha<n$,indomains$\Omega \subseteq$\mathbb{R}$^n$,orRiemannianmanifolds,withpositiveGree{n}^{\mathrm{\prime }}sfunction$G$.