楊明海, 羅慶紅

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一類橢圓邊值問題非平凡解的存在性

楊明海, 羅慶紅

(信陽師范學(xué)院 數(shù)學(xué)與信息科學(xué)學(xué)院, 河南 信陽, 464000)

在一類Ahmad-Lazer-Paul條件下，利用臨界點理論中的廣義山路引理得到了橢圓邊值共振問題非平凡解的存在性.

邊值問題；非平凡解；弱解

本文考慮如下橢圓邊值問題非平凡解的存在性：

定理1的證明需要廣義的山路引理可參見文獻[3]中的定理5.3.

定理1的證明還需要文獻[7]中的引理.

由引理1可知,我們只需證明下面的條件成立：

[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] 楊明海，馮志敏，陳越奮. 半線性橢圓邊值問題解的存在性[J]. 信陽師范學(xué)院學(xué)報: 自然科學(xué)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

信陽師范學(xué)院青年基金資助項目(200959)

楊明海(1980-)，男，講師，博士生，主要從事非線性分析方向的研究.