一类三阶两点边值问题的变号解

摘要

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一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∈ [0,1] subject to u(0) = u'(0) = u"(1) = 0, where f ∈ 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 ∈ [0,1], then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.