Global pointwise estimates of positive solutions to sublinear equations

Abstract:

Bilateral pointwise estimates are provided for positive solutions $u$ to the sublinear integral equation \begin{equation*} u = \mathbf {G}(\sigma u^q) + f \quad \text {in } \ \Omega , \end{equation*} 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 $\Omega \times \Omega$. The main results, which include the existence criteria and uniqueness of solutions, hold true for quasimetric, or quasimetrically modifiable kernels $G$.

As a consequence, bilateral estimates are obtained, along with existence and uniqueness, for positive solutions $u$, possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, \begin{equation*} (-\Delta )^{\frac {\alpha }{2}} u = \sigma u^q + \mu \quad \text {in}\quad \Omega , \quad u=0 \quad \text {in}\,\, \Omega ^c, \end{equation*} where $0<q<1$, and $\mu , \sigma \ge 0$ are measurable functions, or Radon measures, on a bounded uniform domain $\Omega \subset \mathbb {R}^n$ for $0 < \alpha \le 2$, or on the entire space $\mathbb {R}^n$, a ball or half-space, for $0 < \alpha <n$.