Pointwise Estimates of Solutions and Existence Criteria for Sublinear Elliptic Equations

Igor E. Verbitsky · Mathematical Physics and Computer Simulation · 2017

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

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