An a priori estimate for a singly periodic solution of a semilinear equation

Abstract

There exists an exponentially decreasing function $f$ such that any singly $2\pi$-periodic positive solution $u$ of $-\Delta u+u-u^p=0$ in $[0,2\pi ]\times {\mathbb{R}}^{N-1}$ verifies $u(x_1,x^{\prime })\le f(|x^{\prime }|)$. We prove that with the same period and with the same function $f$, any singly periodic positive solution of $-\ep^2Δu-u+u^p=0$ in $[0,2\pi ]\times {\mathbb{R}}^{N-1}$ verifies $u(x_1,x')\leq f(|x'| /\ep )$ . We have a similar estimate for the gradient.

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