一类椭圆边值问题非平凡解的存在性

杨明海, 罗庆红



一类椭圆边值问题非平凡解的存在性

杨明海, 罗庆红

(信阳师范学院 数学与信息科学学院, 河南 信阳, 464000)

在一类Ahmad-Lazer-Paul条件下，利用临界点理论中的广义山路引理得到了椭圆边值共振问题非平凡解的存在性.

边值问题；非平凡解；弱解

本文考虑如下椭圆边值问题非平凡解的存在性：

定理1的证明需要广义的山路引理可参见文献[3]中的定理5.3.

定理1的证明还需要文献[7]中的引理.

由引理1可知,我们只需证明下面的条件成立：

[1] Ahmad S， Lazer A C, Paul J L. Elementary critical point theory and perturbations of elliptic boundary value problems at resonance[J]. Indiana Univ Math, 1976, 25(10): 933-944.

[2] Rabinowitz P. Minimax Methods in Critical Point Theory with Applications to Differential Equations[M]. New York: American Mathematical Society, 1986.

[3] Landesman E M, Lazer A C. Nonlinear perturbations of linear elliptic boundary value problems at resonance[J]. J Math Mech, 1970, 19: 609-623.

[4] Mizoguchi N. Asymptotically Linear Elliptic Equations Without Nonresonance conditions[J]. J Differential Equat- ons, 1994, 113(1): 150-165.

[5] Mawhin J, Ward J R. Nonresonance and existence for nonlinear elliptic boundary value problems[J]. Nonlinear Analysis, 1981, 5(6): 677-684.

[6] Han Z Q. Nontrivial solutions of semilinear elliptic problem via variational methods[J]. Bull Austral Math Soc, 2004, 69(2): 267-275.

[7] 杨明海，冯志敏，陈越奋. 半线性椭圆边值问题解的存在性[J]. 信阳师范学院学报: 自然科学版, 2009, 22(3): 345 -348.

Existence of nontrivial solutions for a class of semilinear elliptic boundary value problems

YANG Ming-hai , LUO Qing-hong

(College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, China)

Existence of nontrivial solutions for a class of semilinear elliptic boundary value resonant problems is obtained under generalized Ahmad-Lazer-Paul type conditions by the generalized mountain pass lemma in the critical point theory.

boundary value problem; nontrivial solution; weak solution

O 175.25

A

1672-6146(2010)03-0001-02

10.3969/j.issn.1672-6146.2010.03.001

2010-07-07

信阳师范学院青年基金资助项目(200959)

杨明海(1980-)，男，讲师，博士生，主要从事非线性分析方向的研究.