馬草川,王亞平,孫小科,裴瑞昌

(1.天水師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,甘肅天水 741001；2.天水市第一中學(xué),甘肅天水 741000)

具有不定位勢(shì)的漸近線性p-Laplacian Dirichlet問題

馬草川1,王亞平2,孫小科1,裴瑞昌1

(1.天水師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,甘肅天水 741001；2.天水市第一中學(xué),甘肅天水 741000)

利用山路引理及極小作用原理,證明了當(dāng)非線性項(xiàng)在無窮遠(yuǎn)處滿足一定的漸近線

性條件時(shí),具有不定位勢(shì)的漸近線性p-Laplacian Dirichlet問題,存在非平凡解.

非平凡解；漸近線性；Dirichlet問題；不定位勢(shì)

1 主要結(jié)果

近年來,具有不定位勢(shì)問題得到了廣泛的研究[1-11],其中文獻(xiàn)[4]考慮了非線性特征值問題

其中Ω是RN(N≥1)中的有界光滑區(qū)域,V(x)滿足條件(2),利用山路引理得到問題(3)的非凡解存在的如下結(jié)果:

引理1[5]若f:Ω×R→R滿足以下假設(shè):時(shí),問題(3)至少有一個(gè)非平凡解.

2 定理的證明

易知u是問題(1)的一個(gè)弱解等價(jià)于I的臨界點(diǎn).

引理2設(shè)e是λ1的非線性特征問題(1)的特征函數(shù),并且定理1中條件成立,則當(dāng)

故由極小作用原理知結(jié)論成立.

[1]Amborosetti A,Rabinowitz P H.Variational methods in critical theory and application[J].Funct.Anal., 1973,14:349-381.

[2]Benci V,Rabinowitz P H.Critical point theorems for indefinite functional[J].Invent.Math.,1979,52:241-273.

[3]Smarandache F.Critical Point Theory and Applications[M].上海:上海科學(xué)技術(shù)出版社,1986.

[4]Cuesta M.Eigenvalue problems for the p-Laplacian with indefinite weight[J],Electronic Journal of Differential Equations,2001,33:1-9.

[5]Xuan B J.Existence results for a super linear p-Laplacian equation with indefinite weights[J].Nonlinear Analysis,2003,54:949-958.

[6]Sun J P,Li W T.Multiple positive solutions to second-order Neumann boundary value problems[J].Appl. Math.Comput.,2003,146:187-194.

[7]Gossez J P,Leadi L.Asymmetric elliptic problems in ?n[J].Electronic Journal of Differential Equations, 2006,14:207-222.

[8]Shao Z Q,Hong J X.The eigenvalue problem for the Laplacian equations[J].Acta.Mathematica Scientia, 2007,02:329-337.

[9]Ariasa M,Camposea J,Cuestab M.An asymmetric Neumann problem with weights[J].Nonlinear Analysis, 2008,25:267-280.

[10]Cuesta M,Quoirin H R.A weighted eigenvalue problem for the p-Laplacian plus a potential[J].Nonlinear Differ.Equ.Appl.,2009,16:469-491.

[11]Dumitru M,Viorica V M,Nikolaos S.Existence and multiplicity of solutions for asymptotically linear, noncoercive elliptic equations[J].Monatsh.Math.,2010,159:59-80.

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.

天水師范學(xué)院中青年教師科研資助項(xiàng)目(TSA0937).

馬草川(1981-),碩士,講師,研究方向：偏微分方程.

2010 MSC:34B15,58E05