Nonlinear potential estimates for sublinear problems with applications to elliptic semilinear and quasilinear equations

2022/10/20 by Verbitsky, Igor E.
#31B15 #35J61 #35J62 (Primary) #42B37 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.11008

Abstract

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type u = G(σu^q) + f \textrmin Ω, where 0 < q < 1, σ≥ 0 is a Radon measure in Ω, f ≥ 0 is a measurable function, and G is a linear integral operator with positive kernel G on Ω×Ω. For quasi-metric (or quasi-metrically modifiable) kernels G, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions u ∈ L^q_{\rm loc} (Ω, σ). Applications are considered to semilinear elliptic equations involving the (fractional) Laplacian, (-Δ)^\fracα2 u = σu^q + μ \textrmin Ω, u=0 \textrmin Ω^c. Here 0

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