一类带有非局部项的四阶椭圆方程无穷多高能量解的存在性

doi:10.6040/j.issn.1671-9352.0.2018.620

《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (6): 81-87.doi: 10.6040/j.issn.1671-9352.0.2018.620

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一类带有非局部项的四阶椭圆方程无穷多高能量解的存在性

张粘,贾高^*

上海理工大学理学院, 上海 200093

发布日期:2019-06-05
作者简介:张粘(1995— ),男,硕士研究生,从事偏微分方程的研究. E-mail:nianzhang411@126.com*通信作者简介:贾高(1960— ),男,博士,教授,从事偏微分方程的研究. E-mail:gaojia89@163.com
基金资助:
国家自然科学基金资助项目(11171220)

Existence of infinitely many high energy solutions of a class of fourth-order elliptic equations with nonlocal terms

ZHANG Nian, JIA Gao^*

School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Published:2019-06-05

摘要/Abstract

摘要： 在全空间上研究了一类带有非局部项的四阶椭圆型方程:{Δ²u-(a+b∫_R^N|∇u|²dx)Δu+V(x)u-1/2Δ(u²)u=f(x,u), x∈R^N,u(x)∈H ²(R^N),其中N≤5,常数a>0, b≥0, Δ²=Δ(Δ)是重调和算子,非线性项f(x,u)不满足AR条件假设,且位势函数V(x)允许变号,利用变分法证明了该类四阶椭圆型方程存在一个高能量的弱解序列。

关键词: Kirchhoff类方程, 变号位势, 变分法, Fountain定理, 无穷多解

Abstract: We study a class of fourth-order elliptic equations with nonlocal term,{Δ²u-(a+b∫_R^N|∇u|²dx)Δu+V(x)u-1/2Δ(u²)u=f(x,u), x∈R^N,u(x)∈H ²(R^N),Where N≤5, constants a>0, b≥0, Δ²=Δ(Δ)is the biharmonic operater, the nonlinearity f(x,u) doesnt satisfy AR condition and the potential function V(x) is also allowed to be sign-changing. We establish the existence of a sequence of high energy weak solutions for this class of elliptic equations via variational methods.

Key words: Kirchhoff type equation, sign-changing potential, variational methods, Fountain theorem, infinitely many solutions

中图分类号:

O175

张粘,贾高. 一类带有非局部项的四阶椭圆方程无穷多高能量解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(6): 81-87.

ZHANG Nian, JIA Gao. Existence of infinitely many high energy solutions of a class of fourth-order elliptic equations with nonlocal terms[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 81-87.

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一类带有非局部项的四阶椭圆方程无穷多高能量解的存在性

Existence of infinitely many high energy solutions of a class of fourth-order elliptic equations with nonlocal terms

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