Uniqueness of positive periodic solutions with some peaks

Abstract

This work deals with the semilinear equation $-\Delta u+u-u^p=0$ in ${\mathbb R}^N$, $2\leq p < {N+2\over N-2}$. We consider the positive solutions, which are ${2\pi\over\varepsilon}$-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 $\varepsilon$ 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 $x_1$, and we estimate the difference between this solution and the groundstate solution.