Pointwise Estimates of Solutions and Existence Criteria for Sublinear Elliptic Equations

Abstract

Abstract. We give a survey of recent results on positive solutions to sublinear elliptic equations of the type − + = , where is an elliptic operator in divergence form, 0 < < 1, ≥ 0 and is a function that may change sign, in a domain Ω ⊆ R , or in a weighted Riemannian manifold, with a positive Green's function . We discuss the existence, as well as global lower and upper pointwise estimates of classical and weak solutions , and conditions that ensure ∈ (Ω) or ∈ 1, (Ω). Some of these results are applicable to homogeneous sublinear integral equations = ( σ) in Ω, where 0 < < 1, and σ = − is a positive locally finite Borel measure in Ω. Here ( σ)( ) = ∫︀ Ω ( , ), ( ) σ( ) is an integral operator with positive (quasi) symmetric kernel on Ω × Ω which satisfies the weak maximum principle. This includes positive solutions, possibly singular, to sublinear equations involving the fractional Laplacian,where 0 < < 1, 0 < α < and = 0 in Ω and at infinity in domains Ω ⊆ R with positive Green's function .

A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

“…Maz'ya and Netrusov [33] (see also [32,Theorem 11.6.1]) gave a capacitary characterization for (1.5). Verbitsky [41] gave another characterization which is based on an inequality of the type (2.3) (see, also [42,36] for related results). More precisely, as in [21, Theorem 1.3], the following two-sided estimate holds:…”