拟线性椭圆型方程Dirichlet问题非平凡解的存在性

THE EXISTENCE OF NONTRIVIAL SOLUTIONS OF THE DIRICHLET PROBLEM FOR QUASILINEAR ELLIPTIC EQUATION

本文利用山路引理在广义Sobolev空间■^(1,F)(Ω)(其中P=(P_1,P_2,…,P_n),P_(?)≥2,i=1,2,…,n)中讨论了下面Dirichlet问题非平凡解的存在性:(?)(x,u,Du)-F_n(x,u,Du)=0,x∈Ω,证明了上述方程在(?)^(1,p)(Ω)中具有非平凡弱解,并且如果I(u)=∫_(Ω)F(x,u,Du)dx是偶泛函,则上述问题具有无穷多个非平凡弱解。

In this paper,we discuss the existence of nontrivial solutions of the following Dirichlet problem in general Sobolev space (Ω)(where P=(P_1,P_2,…,P_n),P_i≥2,i=1,2,…,n)by virtue of the Mountain Pass Lemma: We have proved that the above equation has at least one non- trivial weak solution in (Ω)。And if I(u)=_ΩF(x,u,Du)dx is an even functional then the above problem has infinity many nontrivial weak solutions.