Entire solutions to the Swift--Hohenberg equation via variational approach

Abstract

We study the stationary Swift--Hohenberg equation $(\Delta +1{)}^{2}u-\alpha u-\beta u^2+u^3=0$ in the whole space ${\mathbb{R}}^n$, $2\le n\le 7$. We develop and modify the variational approach introduced by Lerman, Naryshkin and Nazarov (2020) and obtain a series of periodic solutions with certain additional symmetries.

24 pages, 1 figure