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

TL;DR

The paper establishes an exponential decay estimate for positive solutions of a semilinear PDE with periodic boundary conditions, providing bounds that depend on a scaled distance in the unbounded directions.

Contribution

It introduces a priori exponential decay estimates for solutions of a scaled semilinear PDE with periodicity, extending previous bounds to include gradient estimates.

Findings

Solutions are bounded by an exponential function in the unbounded directions.

The bounds scale with the parameter , reflecting the PDE's scaling properties.

Gradient estimates are also established, showing decay similar to the solutions.

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.