Geneviève Allain | Searcharxiv

arXiv2013

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 $x_1$.

math.AP↗

arXiv2011

An a priori estimate for a singly periodic solution of a semilinear equation

There exists an exponentially decreasing function $f$ such that any singly $2π$-periodic positive solution $u$ of $-Δu +u-u^p=0$ in $[0,2π]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'|)$. We prove that with the same period and with the same function $f$, any singly periodic positive solution of $-\ep^2Δu-u+u^p=0$ in $[0,2π]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'| /\ep )$ . We have a similar estimate for the gradient.

math.AP↗