Global pointwise estimates of positive solutions to sublinear equations

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation $u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω,$ for $0 < q < 1$, where $σ\ge 0$ 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 $Ω\timesΩ$. 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, $(-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c,$ where $0<q<1$, and $μ, σ\ge 0$ are measurable functions, or Radon measures, on a bounded uniform domain $Ω\subset \mathbf{R}^n$ for $0 < α\le 2$, or on the entire space $\mathbf{R}^n$, a ball or half-space, for $0 < α<n$.

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