An approximate Gidas–Ni–Nirenberg theorem

We study positive solutions u to Δ u + f ( u ) = 0 in Ω, u = 0 on ∂Ω, and we address the following question: If Ω is a small perturbation of a ball, is u a small perturbation of a radially symmetric function? We prove two theorems which give an affirmative answer under different assumptions on the non‐linearity f and on the topologies in which perturbations are considered.