吕定洋



一类四阶椭圆型方程的高能解的存在性

吕定洋

（湖南第一师范学院数学与计算科学学院，湖南，长沙 410205）

研究了一类四阶椭圆型方程的高能解的存在性，在对非线性项作新的假设条件下，利用临界点理论得到了方程无穷多高能解的存在性结果，对非线性项所作的假设比已有文献的假设要弱。

四阶椭圆型方程；临界点理论；对称山路定理；高能解

0 引言

考虑如下非线性四阶椭圆型方程

已经有很多学者研究了方程(1.1)，见文献[1-3]。在文献[1]中，Yin和Wu研究了方程(1.1)，得到了(1.1)高能解的两个存在性结果。在文献[2]中，Ye和Tang在如下的假设下，得到了(1.1)高能解的存在性结论：

1　变分框架和主要引理

为了寻找问题(1.1)的弱解，将考虑如下定义在上的泛函：

在上面的假设下，易知，且

2　主要结果的证明

结合(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 Schrödinger 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 Schrödinger 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

吕定洋(1968-)，男，湖南邵阳人，副教授，硕士，主要从事偏微分方程临界点理论研究(E-mail:hncsydy@126.com).