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

We consider the positive solutions u of −∆u+u−u = 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 −∆v + v − v = 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.