二维Landau－Lifshitz静态方程组

LANDAU-LIFSHITZ EQUILIBRIUM SYSTEMS IN TWO DIMENSIONS

本文讨论了下述边值问题我们证明了当时，上述问题的极小解存在．当ｎ＝２，ｕ０＝（０，０，１）且当λ≤０时，ｕ＝ｕ０是唯一正则解；当０＜λ≤λ１时，除了ｕ＝ｕ０。是唯一的能量极小解外，还存在一个非常数的解；当λ＞λ１时，ｕ＝ｕ０是一个非极小解，并至少存在两个非常数解．

This paper deals with the following boundary value problem'We prove that there exists a mlruxmzing solution for the above problem fgr any bounded domain D in Rn, n≥2. When , we get the following results' (i) if λ≤0,then u=uo is the unique solution; (ii) if 0<λ≤λ1, there exists a nonconstant solution, in addition to the minimizing solution u=u0; (iii) if λ>λ1, u=u0 is not mining and thereexist at least two nonconstant solutions.