具有非线性边界和渐近线性非线性项的椭圆方程的多解存在性  

Solutions to an Asymptotic Linear Elliptic Problem With Nonlinear Boundary Condition

作　　者：董旺远[1] 王征平[2] 

机构地区：[1]华中农业大学理学院,武汉湖北430070 [2]中国科学院武汉物理与数学研究所,武汉湖北430071

出　　处：《数学进展》2009年第4期481-487,共7页Advances in Mathematics（China）

基　　金：supported by the Doctoral Science Foundation of HZAU(No.52204-07119);the second author is supported by NSFC(No.10571174).

摘　　要：本文研究下面的椭圆方程:其中Ω是R^N(N≥3)中的光滑有界区域,1<q<2,λ是一个正参数,f(x,u)是渐近线性的。应用Ekeland变分原理和山路定理,我们证明当λ充分小时,方程至少有两个非平凡解。In this paper we study the existence of nontrivial solutions to the problem {-△u（x）＋u（x）=f（x,u）,x∈Ω, u/ v=λ｜u｜^q-2u,x∈ Ω, where Ω is a bounded smooth domain in R^N（N 〉 3）, 1 〈 q 〈 2, λ is a positive parameter and f（x, s） is asymptotically linear with respect to s at infinity. By Ekeland＇s variational principle and a variant version of Mountain Pass Theorem, we prove that there exist at least two nontrivial solutions if λ is small.

关 键 词：渐近线性 非线性边界 EKELAND变分原理 山路定理 

分 类 号：O175.23[理学—数学]