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We give a survey of recent results on positive solutions to sublinear elliptic equations of the type −Lu+V u = 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, 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(Ω) or u ∈ W (Ω). Some of these results are applicable to homogeneous sublinear integral equations u = G(udσ) 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, (−Δ) α 2 u = σu, u ≥ 0 in Ω, where 0 < q < 1, 0 < α < n and u = 0 in Ω and at infinity in domains Ω ⊆ R with positive Green’s function G.

Pointwise Estimates of Solutions and Existence Criteria for Sublinear Elliptic Equations