A mean field equation on a torus: one-dimensional symmetry of solutions

Abstract

We study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and $\oep$ is a rectangle of $\R^2$ with side lengths $1/\epsilon$ and $1$, $0< \epsilon \leq 1$. We establish that every solution depends only on the $x$--variable when $\lambda \leq \lambda^*(\epsilon)$, where $\lambda^*(\epsilon)$ is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters $\epsilon$ and $\lambda$ in which every solution is identically zero. This range is optimal for $\epsilon\leq1/2$.