含一阶导数的半线性四阶边值问题的多重正解

Multiple Positive Solutions of a Semilinear Fourth-Order Boundary Value Problem with First Derivative

通过构造适当的锥并且利用方程的分解技巧研究了一类含一阶导数的半线性四阶边值问题的正解.主要工具是三阶两点边值问题的一个Green函数及锥拉伸与锥压缩型的Krasnasel’skii不动点定理.在力学中,这类问题描述了一端固定,另一端活动的弹性梁的形变.结论表明只要非线性项在某些有界集合上的“高度”适当,这类问题至少存在n个正解.

By constructing suitable cone and applying the decomposition technique of equation, the positive solutions are investigated for a class of semilinear fourth-order boundary value problems with first derivative. Main tools are a Green function of the third-order two-point boundary value problems and Krasnosel＇skii fixed point theorem of cone expansion-compression type. In the mechanics, the class of problems describes the deformation of an elastic beam whose one end is fixed and other is movable. Our results show that the problem has at least n positive solutions provided the ＂heights＂ of nonlinear term are appropriate on some bounded sets.