Singular set of solutions of semilinear elliptic equations

Abstract

We study the behavior of the singular set {u=∇u=0} for solutions u to the semilinear elliptic equation Δu=f(x,u,∇u),x∈Ω, where Ω is an open set in Rn and |f(x,u,∇u)|≤A|u|p+B|∇u|q, where A,B≥0 and p,q∈(0,∞). We show that in dimension n=2 the singular set is a discrete set, and if min{p,q}<1 then a solution u satisfies an optimal growth βp,q near every non-isolated point of the singular set. Also the same results are true in dimension n≥3 under an (n−1)-dimensional density condition.