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

We consider the positive solutions u of −Δu+u−u^p=0 in [0,2π]×R^N−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−v^p=0 in R^N−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.