一类奇异梁方程三个正解的存在性

Existence of the Triple Positive Solutions to a Singular Beam Equation

本文利用格林函数方法和Leggett-Williams不动点定理,讨论了一类非线性四阶两点奇异边值问题多个正解的存在性问题,得出当非线性项满足一定条件时,该边值问题至少有三个正解存在.在力学上,该模型模拟了左端简单支撑右端被滑动夹子夹住的弹性梁的挠曲,由于非线性项中涉及弯矩,因此主要结论对于梁的稳定性分析是有益的.最后,数值算例进一步证实本文理论及方法的严密性和有效性.

By applying the technique of Green function and the Leggett-Williams fixed point theo-rem,we study the existence of multiple positive solutions for a class of nonlinear fourth-order two-point boundary value problems with singularity,and conclude that if the nonlinear term satisfies some con-ditions,the boundary value problems have at least three positive solutions.In mechanics,the model describes the bending of an elastic beam which is simply supported at left and clamped at right by sliding clamps.Since the nonlinear term involves bending argument,the results are useful for the stability analysis of the beam.Finally,numerical example indicates the strictness and validity of our theorem and method.