## Advances in Differential Equations

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

### Uniqueness of positive periodic solutions with some peaks

Geneviève Allain and Anne Beaulieu

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 November 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1384278132

**Mathematical Reviews number (MathSciNet)**

MR3161656

**Zentralblatt MATH identifier**

1286.35019

**Subjects**

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

#### Citation

Allain, Geneviève; Beaulieu, Anne. Uniqueness of positive periodic solutions with some peaks. Adv. Differential Equations 19 (2014), no. 1/2, 51--86.https://projecteuclid.org/euclid.ade/1384278132