楊明海, 羅慶紅

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一類(lèi)橢圓邊值問(wèn)題非平凡解的存在性

楊明海, 羅慶紅

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

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

邊值問(wèn)題；非平凡解；弱解

本文考慮如下橢圓邊值問(wèn)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(jiàn)文獻(xiàn)[3]中的定理5.3.

定理1的證明還需要文獻(xiàn)[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(xiàn)性橢圓邊值問(wèn)題解的存在性[J]. 信陽(yáng)師范學(xué)院學(xué)報(bào): 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(yáng)師范學(xué)院青年基金資助項(xiàng)目(200959)

楊明海(1980-)，男，講師，博士生，主要從事非線(xiàn)性分析方向的研究.