Uniqueness of positive periodic solutions with some peaks

TL;DR

This paper investigates the uniqueness and distribution of positive periodic solutions with peaks for a semi-linear elliptic equation, showing that peak points must be asymptotically uniformly spaced and establishing uniqueness for small periods.

Contribution

It proves the asymptotic uniform distribution of peaks and the uniqueness (up to translation) of solutions with specified peaks for small periods in a semi-linear elliptic PDE.

Findings

Peak points are asymptotically uniformly distributed.

Uniqueness of solutions with given peaks for small periods.

Solutions decay to zero away from peaks.

Abstract

This work deals with the semi linear equation $-\Delta u+u-u^p=0$ in $\R^N$, $2\leq p<{N+2\over N-2}$. We consider the positive solutions which are ${2\pi\over\ep}$-periodic in $x_1$ and decreasing to 0 in the other variables, uniformly in $x_1$. Let a periodic configuration of points be given on the $x_1$-axis, which repel each other as the period tends to infinity. If there exists a solution which has these points as peaks, we prove that the points must be asymptotically uniformly distributed on the $x_1$-axis. Then, for $\ep$ small enough, we prove the uniqueness up to a translation of the positive solution with some peaks on the $x_1$-axis, for a given minimal period in $x_1$.