具有不定位势的渐近线性p-Laplacian Dirichlet问题

马草川,王亚平,孙小科,裴瑞昌

(1.天水师范学院数学与统计学院,甘肃天水 741001；2.天水市第一中学,甘肃天水 741000)

具有不定位势的渐近线性p-Laplacian Dirichlet问题

马草川1,王亚平2,孙小科1,裴瑞昌1

(1.天水师范学院数学与统计学院,甘肃天水 741001；2.天水市第一中学,甘肃天水 741000)

利用山路引理及极小作用原理,证明了当非线性项在无穷远处满足一定的渐近线

性条件时,具有不定位势的渐近线性p-Laplacian Dirichlet问题,存在非平凡解.

非平凡解；渐近线性；Dirichlet问题；不定位势

1 主要结果

近年来,具有不定位势问题得到了广泛的研究[1-11],其中文献[4]考虑了非线性特征值问题

其中Ω是RN(N≥1)中的有界光滑区域,V(x)满足条件(2),利用山路引理得到问题(3)的非凡解存在的如下结果:

引理1[5]若f:Ω×R→R满足以下假设:时,问题(3)至少有一个非平凡解.

2 定理的证明

易知u是问题(1)的一个弱解等价于I的临界点.

引理2设e是λ1的非线性特征问题(1)的特征函数,并且定理1中条件成立,则当

故由极小作用原理知结论成立.

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Asymptotically linear p-Laplacian Dirichlet problem with
indefinite weights

Ma Caochuan1,Wang Yaping2,Sun Xiaoke1,Pei Ruichang1

(1.School of Mathematics and Statistics,Tianshui Normal University,Tianshui741001,China； 2.Tianshui No.1 Middle School,Tianshui741000,China)

By using mountain pass and the least action theorems,the existience of nontrivial solution is obtained for a class of asymptotically linear p-Laplacian Dirichlet problem with indefinite weights.

nontrivial solution,asymptotically linear,Dirichlet problem,indefinite weights

O175. 23；O176.3

A

1008-5513(2012)04-0501-06

2011-07-02.

天水师范学院中青年教师科研资助项目(TSA0937).

马草川(1981-),硕士,讲师,研究方向：偏微分方程.

2010 MSC:34B15,58E05