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

We study the equation [GRAPHICS] for u is an element of E , where E = {u is an element of H-1 (Omega(epsilon)): u is doubly periodic, integral(Omega epsilon) u = 0} and Omega(epsilon) is a rectangle of R-2 with side lengths 1/epsilon and 1, 0 < epsilon <= 1. We establish that every solution depends only on the x -variable when lambda <= 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 <= 1/2.