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.
A uniqueness result for a semilinear elliptic problem: A computer-assisted proof / P. J., Mckenna; Pacella, Filomena; M., Plum; D., Roth. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 247:7(2009), pp. 2140-2162. [10.1016/j.jde.2009.06.023]
A uniqueness result for a semilinear elliptic problem: A computer-assisted proof
PACELLA, Filomena;
2009
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 -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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.