呂定洋

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一類四階橢圓型方程的高能解的存在性

呂定洋

（湖南第一師范學院數(shù)學與計算科學學院，湖南，長沙 410205）

研究了一類四階橢圓型方程的高能解的存在性，在對非線性項作新的假設(shè)條件下，利用臨界點理論得到了方程無窮多高能解的存在性結(jié)果，對非線性項所作的假設(shè)比已有文獻的假設(shè)要弱。

四階橢圓型方程；臨界點理論；對稱山路定理；高能解

0 引言

考慮如下非線性四階橢圓型方程

已經(jīng)有很多學者研究了方程(1.1)，見文獻[1-3]。在文獻[1]中，Yin和Wu研究了方程(1.1)，得到了(1.1)高能解的兩個存在性結(jié)果。在文獻[2]中，Ye和Tang在如下的假設(shè)下，得到了(1.1)高能解的存在性結(jié)論：

1　變分框架和主要引理

為了尋找問題(1.1)的弱解，將考慮如下定義在上的泛函：

在上面的假設(shè)下，易知，且

2　主要結(jié)果的證明

結(jié)合(2)，(3)可知

[1] Yin Y, Wu X. High energy solutions and nontrivial solutions for fourth-order elliptic equations[J]. Journal of Mathematical Analysis and Applications,2011,375(2): 699-705.

[2] Ye Y, Tang C L. Infinitely many solutions for fourth-order elliptic equations[J]. Journal of Mathematical Analysis and Applications, 2012, 394(2): 841-854.

[3] Liu J, Chen S X, Wu X. Existence and multiplicity of solutions for a class of fourth-order elliptic equations in RN[J]. Journal of Mathematical Analysis and Applications, 2012, 395(2): 608-615.

[4] Zhang Q, Xu B. Multiplicity of solutions for a class of semilinear Schr?dinger equations with sign-changing potential[J]. Journal of Mathematical Analysis and Applications, 2011, 377(2): 834-840.

[5] Tang X H. Infinitely many solutions for semilinear Schr?dinger equations with sign-changing potential and nonlinearity[J]. Journal of Mathematical Analysis and Applications, 2013, 401(1): 407-415.

[6] Zhang W, Tang X, Zhang J. Infinitely many solutions for fourth-order elliptic equations with sign-changing potential[J]. Taiwanese Journal of Mathematics, 2014, 18(2): 645-659.

EXISTENCE OF INFINITELY MANY LARGE SOLUTIONS FOR THE FOURTH-ORDER ELLIPTIC EQUATIONS

LV Ding-yang

(Department of Mathematics ,Hunan First Normal University, Changsha, Hunan 410205, China)

We study the existence of infinitely many high energy solutions for a class of fourth-order elliptic equations. Under new assumptions on nonlinearity, by using critical point theory, we obtain infinitely many large solutions for the equations. our assumptions on nonlinearityare weaker than the existing results.

fourth-order ellipticequations; critical point theory; symmetric mountain pass theorem; high energy solutions

1674-8085(2018)03-0001-04

O175

A

10.3969/j.issn.1674-8085.2018.03.001

2018-03-07；

2018-04-16

呂定洋(1968-)，男，湖南邵陽人，副教授，碩士，主要從事偏微分方程臨界點理論研究(E-mail:hncsydy@126.com).