退缩拟线性椭圆型方程某类边值问题的无穷多解

INFINITELY MANY SOLUTIONS OF CERTAIN BOUNDARY VALUE PROBLEM FOR QUASILINEAR DEGENERATE ELLIPTIC EQUATION

文[1]证明了如下D氏问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, u=0,x∈Ω存在非平凡解,本文讨论上述方程的另一类边界问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, g(|Du|~2)D_iu(0)(n,x_i)+h(x,u)=0,x∈Ω, (1)其中Ω∈R^n是具有光滑边界的有界区域,n(x)是Ω在x点的外法向,D_iu=u/x_i,Du=gradu=u,重复指标表示求和,与问题(1)相应的泛函为:

In this paper, we prove that the problemhas infinitely many nontrivial generalized solutions, where g(0) = 0,Ω Rn being a bounded smooth domain, n(x) being a unit outward normal to Ω .