Singular set of solutions of semilinear elliptic equations

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