一类三阶两点边值问题的变号解

Sign-changing Solutions to Third-order Two-point Boundary Value Problem

利用不动点指数理论和Leray-Schauder度理论讨论带有边值u(0)=u′(0)=u″(1)=0的三阶两点边值问题-u″′(t)=f(t,u(t)),t∈[0,1],其中f∈C([0,1]×R,R).通过计算相应的线性算子的特征值与代数重数,获得了一些包括变号解的存在性结果.如果f满足一定的条件,则问题至少存在六个不同的非平凡解,其中两个正解,两个负解以及两个变号解.进一步,如果f(t,·),t∈[0,1]是奇函数,则问题至少存在八个不同的非平凡解,其中两个正解,两个负解以及四个变号解.

In this paper, we use the fixed point index theory and the Leray-Schauder degree theory to discuss the third-order boundary value problem -u＇＂（t） = f（t, u（t）） for all t E [0, 1] subject to u（0） = u＇（0） = u＂（1） = 0, where f E C（[0,1] ~ R,R）. By computing hardly the eigenvalues and their algebraic multiplicities of the associated linear problem, we obtain some new existence results concerning sign-changing solutions to this problem. If f satisfies certain conditions, then the problem has at least six different nontrivial solutions： two positive solutions, two negative solutions and two sign-changing solutions. Moreover, if f（t, .） is odd for all t E [0, 1], then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.