Global pointwise estimates of positive solutions to sublinear equations

TL;DR

This paper establishes bilateral pointwise estimates, existence, and uniqueness criteria for positive solutions to sublinear integral and elliptic equations involving fractional Laplacians, applicable to various domains and kernel types.

Contribution

It provides new bilateral estimates and criteria for existence and uniqueness of solutions to sublinear equations with fractional Laplacians, extending to unbounded solutions and general kernels.

Findings

Bilateral pointwise estimates for solutions

Existence and uniqueness criteria established

Applicable to fractional Laplacian equations in various domains

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 $0 < \alpha <n$.