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Uniqueness of positive periodic solutions with some peaks

This work deals with the semi linear equation $-Δu+u-u^p=0$ in $\R^N$, $2\leq p<{N+2\over N-2}$. We consider the positive solutions which are ${2π\over\ep}$-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 $x_1$.