＊ 一类四阶边值问题的变号解

李宇华

(山西大学数学科学学院,山西太原 030006)

＊一类四阶边值问题的变号解

李宇华

(山西大学数学科学学院,山西太原 030006)

利用拓扑度理论和Mo rse理论研究方程u(4)(t)=f(t,u),t∈(0,1),且带有边界条件u″(0)=u″(1)=0,u(0)=u(1)=0.在一定条件下,得到此问题有六个解,其中两个正解,两个负解,两个变号解.

临界群;变号解;Mo rse理论

本文研究以下四阶方程

目前已经有许多文章研究四阶边值问题(1),见文献[1-3],例如,利用锥拉伸和锥压缩不动点定理,得到其正解的存在性,见文[1].利用临界点理论得到非平凡解的存在性,见文献[2],文献[3]利用拓扑度理论研究了变号解的存在性.然而他们没有考虑共振情形,也没有考虑f0(t)和f∞(t)不等于常数的情形,本文把拓扑度理论和Morse理论结合起来,考虑了各种情形下问题(1)的变号解的存在性与多重性,把跨特征值和共振情形统一起来,推广了文献[3]的结果.这是仅仅利用拓扑度理论或Morse理论都无法得到的.本文假设f满足以下条件:

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Sign-changing Solutions to Fourth-order Boundary Value Problem

LI Yu-hua
(School of Mathematical Sciences,Shanxi University,Taiyuan030006,China)

The problemu(4)(t)=f(t,u),t∈(0,1)with the boundary value conditionsu″(0)=u″(1)=0,u(0)=u(1)=0 is studied by using topological degree and Morse theory.Under some conditions,w e obtain this problem has at least six solutions,including two positive solutions,two negative solutions and two sign-changing solutions.

critical group;sign-changing solutions;Morse theory

O152.7

A

0253-2395(2011)01-0001-04＊

2010-08-05;

2010-09-07

国家自然科学基金(10771128;11071149);山西省自然科学基金(2006011002;20100110011)

李宇华(1981-),女,山西五台人,讲师,在读博士.E-mail:yhli@sxu.edu.cn