＊ 一類(lèi)四階邊值問(wèn)題的變號(hào)解

李宇華

(山西大學(xué)數(shù)學(xué)科學(xué)學(xué)院,山西太原 030006)

＊一類(lèi)四階邊值問(wèn)題的變號(hào)解

李宇華

(山西大學(xué)數(shù)學(xué)科學(xué)學(xué)院,山西太原 030006)

利用拓?fù)涠壤碚摵蚆o rse理論研究方程u(4)(t)=f(t,u),t∈(0,1),且?guī)в羞吔鐥l件u″(0)=u″(1)=0,u(0)=u(1)=0.在一定條件下,得到此問(wèn)題有六個(gè)解,其中兩個(gè)正解,兩個(gè)負(fù)解,兩個(gè)變號(hào)解.

臨界群;變號(hào)解;Mo rse理論

本文研究以下四階方程

目前已經(jīng)有許多文章研究四階邊值問(wèn)題(1),見(jiàn)文獻(xiàn)[1-3],例如,利用錐拉伸和錐壓縮不動(dòng)點(diǎn)定理,得到其正解的存在性,見(jiàn)文[1].利用臨界點(diǎn)理論得到非平凡解的存在性,見(jiàn)文獻(xiàn)[2],文獻(xiàn)[3]利用拓?fù)涠壤碚撗芯苛俗兲?hào)解的存在性.然而他們沒(méi)有考慮共振情形,也沒(méi)有考慮f0(t)和f∞(t)不等于常數(shù)的情形,本文把拓?fù)涠壤碚摵蚆orse理論結(jié)合起來(lái),考慮了各種情形下問(wèn)題(1)的變號(hào)解的存在性與多重性,把跨特征值和共振情形統(tǒng)一起來(lái),推廣了文獻(xiàn)[3]的結(jié)果.這是僅僅利用拓?fù)涠壤碚摶騇orse理論都無(wú)法得到的.本文假設(shè)f滿(mǎn)足以下條件:

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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

國(guó)家自然科學(xué)基金(10771128;11071149);山西省自然科學(xué)基金(2006011002;20100110011)

李宇華(1981-),女,山西五臺(tái)人,講師,在讀博士.E-mail:yhli@sxu.edu.cn