# Uniqueness of positive periodic solutions with some 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 $x_1$.
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Preprints, Working Papers, ...

https://hal-upec-upem.archives-ouvertes.fr/hal-00804269
Contributor : Anne Beaulieu <>
Submitted on : Monday, March 25, 2013 - 1:58:09 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM
Long-term archiving on: : Wednesday, June 26, 2013 - 4:01:49 AM

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Uniqueness-mars-2013.pdf
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### Identifiers

• HAL Id : hal-00804269, version 1
• ARXIV : 1303.6139

### Citation

Geneviève Allain, Anne Beaulieu. Uniqueness of positive periodic solutions with some peaks.. 2013. ⟨hal-00804269⟩

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