A sublinear version of Schur’s lemma and elliptic PDE

along with its weak-type analogue, for $0 < q < 1$ , where $G$ is an integral operator associated with the nonnegative kernel $G$ on $Ω \times Ω$ . Here $ℳ^{+} (Ω)$ denotes the class of positive Radon measures in $Ω$ ; $σ, ν \in ℳ^{+} (Ω)$ , and $∥ ν ∥ = ν (Ω)$ .

For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $σ$ for which such an embedding holds. The strong-type $(1, q)$ -inequality for $0 < q < 1$ is closely connected with existence of a positive function $u$ such that $u \geq G (u^{q} σ)$ , i.e., a supersolution to the integral equation

u - G (u^{q} σ) = 0, u \in L_{l o c}^{q} (Ω, σ) .

This study is motivated by solving sublinear equations involving the fractional Laplacian,

{(- Δ)}^{\frac{α}{2}} u - u^{q} σ = 0,

in domains $Ω \subseteq ℝ^{n}$ which have a positive Green function $G$ for $0 < α < n$ .

Keywords

weighted norm inequalities, sublinear elliptic equations, Green's function, weak maximum principle, fractional Laplacian

Mathematical Subject Classification 2010

Primary: 35J61, 42B37

Secondary: 31B15, 42B25

Milestones

Received: 10 February 2017

Revised: 14 July 2017

Accepted: 5 September 2017

Published: 17 October 2017

Authors

Stephen Quinn
Department of Mathematics University of Missouri Columbia, MO United States

Igor E. Verbitsky
Department of Mathematics University of Missouri Columbia, MO United States

Vol. 11, No. 2, 2018

Stephen Quinn and Igor E. Verbitsky

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors