Affiliations: Laboratoire d'Analyse et de Mathématiques Appliquées, Faculté de Sciences et Technologie, Université Paris-Est Créteil, Créteil, France | Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris-Est Marne la Vallée, Marne la Vallée, France
Note: [] Corresponding author: Anne Beaulieu, Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris-Est Marne la Vallée, UMR CNRS 8050, 5 boulevard Descartes, 77454 Marne la Vallée cedex 2, France. E-mail: anne.beaulieu@univ-mlv.fr.
Abstract: We consider the positive solutions u of −Δu+u−up=0 in [0,2π]×RN−1, which are 2π-periodic in x1 and tend uniformly to 0 in the other variables. There exists a constant C such that any solution u verifies u(x1,x′)≤Cw0(x′) where w0 is the ground state solution of −Δv+v−vp=0 in RN−1. We prove that exactly the same estimate is true when the period is 2π/ε, even when ε tends to 0. We have a similar result for the gradient.
Keywords: asymptotic behavior of solutions, semilinear elliptic equations, periodic solutions
