Advances in Differential Equations

Uniqueness of positive periodic solutions with some peaks

Geneviève Allain and Anne Beaulieu

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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.

Article information

Adv. Differential Equations, Volume 19, Number 1/2 (2014), 51-86.

First available in Project Euclid: 12 November 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B09: Positive solutions 35B10: Periodic solutions 35B40: Asymptotic behavior of solutions


Allain, Geneviève; Beaulieu, Anne. Uniqueness of positive periodic solutions with some peaks. Adv. Differential Equations 19 (2014), no. 1/2, 51--86.

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