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

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 -Delta u = lambda u + u(p) in Omega, u = 0 on partial derivative Omega, where p > 1 and lambda ranges between 0 and the first Dirichlet eigenvalue lambda(1)(Omega) of -Delta. For the case when Omega is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Omega is not a ball, and then only for lambda = 0. In this article, we prove uniqueness, for all lambda is an element of [0, lambda(1)(Omega)), in the case Omega = (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 lambda lose to lambda(1)(Omega), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem. (C) 2009 Elsevier Inc. All rights reserved