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

作者：Cabre X^*; Lucia M; Sanchon M

来源：Communications in Partial Differential Equations, 2005, 30(9): 1315-1330.

DOI：10.1080/0360530050028857

• Bol's inequality
• conformal radius
• mean field equations
• one-dimensional symmetry
• periodic solutions

We study the equation
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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.

• 出版日期2005