摘要
讨论了有界区域上的Dirichlet问题-△u-λu=α(x)│u│^(p-1)u+f(x,u),x∈Ω,u=0,x∈Ω非平凡解的存在性。其中 p=(n+2)/(n-2),n≥3,f(x,u)是关于│u│的增涨阶低于p的连续函数,λ是正参数。我们先证明了一个不具(PS)条件的临界点定理。据此并利用Sobolev嵌入定理的最优常数,克服了失去紧性的困难,从而得到非平凡解的存在性。与Brezis—Nirenberg结果不同的是,我们没有假设λ<λ_1,λ_1是-△:H_0~1(Ω)→H^(-1)(Ω)的第一本征值。
The existence of nontrivial solutions of the Dirichlet - △u-λu = a(x) |u|p-1u + f(x,u) in Ω,u = 0 on Ωis studied,in which p =n+2/n-2,n≥3,f(x,u) is a lower - order
perturbation of \u\p in the sense that = lim f(x,u)/ |u|p=0,λ>0 isa real parameter. First a
critical point theorem without the (PS) condition is proved. The problem of the lack of compactness is solved. In our main results we don't suppose λ< λ1 which is necessary for the relative results of Brezis - Nirenberg,where λ1 is the first eigenvalue of the operator- △:H01(Ω)→H-1(Ω).
出处
《重庆大学学报（自然科学版）》
EI
CAS
CSCD
1991年第2期23-31,共9页
Journal of Chongqing University
关键词
非线性
椭圆方程
非平凡解
变分法
variational method
nonlinear elliptic equation/nontrivial solution
critical Sobolev exponent
