Global pointwise estimates of positive solutions to sublinear equations

Abstract

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω,\] 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, \[ (-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c, \] 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 $0<\alpha <n$.

34 pages