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An a priori estimate for the singly periodic solutions of a semilinear equation

Abstract : We consider the positive solutions u of -Delta u + u - u(p) = 0 in [ 0,2 pi] x RN - 1, which are 2 pi-periodic in x(1) and tend uniformly to 0 in the other variables. There exists a constant C such that any solution u verifies u( x(1), x(1)) <= Cw(0)(x(1)) where w(0) is the ground state solution of -Delta v + v - v(p) = 0 in RN - 1. We prove that exactly the same estimate is true when the period is 2 pi/epsilon, even when epsilon tends to 0. We have a similar result for the gradient.
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Submitted on : Tuesday, May 1, 2012 - 12:14:00 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

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Genevieve Allain, Anne Beaulieu. An a priori estimate for the singly periodic solutions of a semilinear equation. Asymptotic Analysis, IOS Press, 2012, 76 (2), pp.115--122. ⟨10.3233/ASY-2011-1076⟩. ⟨hal-00692702⟩

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