A uniqueness result for a semilinear elliptic problem: A computer-assisted proof

Abstract Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209–243], many papers have been devoted to the uniqueness question for positive solutions of − Δ u = λ u + u p in Ω, u = 0 on ∂Ω, where p > 1 and λ ranges between 0 and the first Dirichlet eigenvalue λ 1 ( Ω ) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ = 0 . In this article, we prove uniqueness, for all λ ∈ [ 0 , λ 1 ( Ω ) ) , in the case Ω = ( 0 , 1 ) 2 and p = 2 . This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ 1 ( Ω ) , we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.