Singly periodic solutions of a semilinear equation. - Laboratoire d'Analyse et de Mathématiques Appliquées Accéder directement au contenu
Article Dans Une Revue Annales de l'Institut Henri Poincaré C, Analyse non linéaire Année : 2009

Singly periodic solutions of a semilinear equation.

Résumé

We consider the solutions of the equation −ε2Δu+u−|u|p−1u=0 in S1×R, where ε and p are positive real numbers, p>1. We prove that the set of the positive bounded solutions even in x1 and x2, decreasing for x1∈]−π,0[ and tending to 0 as x2 tends to +∞ is the first branch of solutions constructed by bifurcation from the ground-state solution. We prove that there exists a positive real number ε⋆ such that for every ε∈]0,ε⋆] there exists a finite number of solutions verifying the above properties and none such solution for ε>ε⋆. The proves make use of compactness results and of the Leray-Schauder degree theory.

Dates et versions

hal-00727776 , version 1 (04-09-2012)

Identifiants

Citer

Geneviève Allain, Anne Beaulieu. Singly periodic solutions of a semilinear equation.. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2009, 26 (4), pp.1277-1297. ⟨10.1016/j.anihpc.2008.10.001⟩. ⟨hal-00727776⟩
86 Consultations
1 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More