Global pointwise estimates of positive solutions to sublinear equations

Abstract

Bilateral pointwise estimates are provided for positive solutions u u to the sublinear integral equation u = G ( σ u q ) + f in    Ω , \begin{equation*} u = \mathbf {G}(\sigma u^q) + f \quad \text {in } \ \Omega , \end{equation*} for 0 > q > 1 0 > q > 1 , where σ ≥ 0 \sigma \ge 0 is a measurable function or a Radon measure, f ≥ 0 f \ge 0 , and G \mathbf {G} is the integral operator associated with a positive kernel G G on Ω × Ω \Omega \times \Omega . The main results, which include the existence criteria and uniqueness of solutions, hold true for quasimetric, or quasimetrically modifiable kernels  G G .

As a consequence, bilateral estimates are obtained, along with existence and uniqueness, for positive solutions u u , possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, ( − Δ ) α 2 u = σ u q + μ in Ω , u = 0 in Ω c , \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 0>q>1 , and μ , σ ≥ 0 \mu , \sigma \ge 0 are measurable functions, or Radon measures, on a bounded uniform domain Ω ⊂ R n \Omega \subset \mathbb {R}^n for 0 > α ≤ 2 0 > \alpha \le 2 , or on the entire space R n \mathbb {R}^n , a ball or half-space, for 0 > α > n 0 > \alpha >n .