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