  We give bilateral pointwise estimates for positive solutions $u$ to the
sublinear integral equation
  \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega,\] for $0 < q
< 1$, where $\sigma\ge 0$ is a measurable function, or a Radon measure, $f \ge
0$, and $\mathbf{G}$ is the integral operator associated with a positive kernel
$G$ on $\Omega\times\Omega$. Our main results, which include the existence
criteria and uniqueness of solutions, hold for quasi-metric, or
quasi-metrically modifiable kernels $G$.
  As a consequence, we obtain bilateral estimates, along with the existence and
uniqueness, for positive solutions $u$, possibly unbounded, to sublinear
elliptic equations involving the fractional Laplacian,
  \[ (-\Delta)^{\frac{\alpha}{2}} u = \sigma u^q + \mu \quad \textrm{in} \,\,
\Omega, \qquad u=0 \, \, \textrm{in} \,\, \Omega^c, \] where $0<q<1$, and $\mu,
\sigma \ge 0$ are measurable functions, or Radon measures, on a bounded uniform
domain $\Omega \subset \mathbf{R}^n$ for $0 < \alpha \le 2$, or on the entire
space $\mathbf{R}^n$, a ball or half-space, for $0 < \alpha <n$.

Analysis of PDEs