Global pointwise estimates of positive solutions to sublinear equations

Abstract

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 $0 < \alpha <n$.