允许 Green 函数取零值情形下 Neumann 问题的正解

Positive solutions of Neumann problem with Green's function vanishing at some points

摘要:

本文研究了一类二阶非线性常微分方程Neumann边值问题 $$ \left\{\begin{array}{ll} y''+\ a(t)y=\lambda g(t)f(y),~~\ \ \ t\in [0,1],\\[2ex] \ y'(0)=\ y'(1)=0, \end{array} \right.\eqno $$正解的存在性,~其中~$\lambda$~是一个正参数,~$f$~在~$\infty$~处是超线性的且~$f$ 允许变号,此外与这一问题相关的Green 函数可以在某些点等于0. 主要结果的证明基于Krasnosel'skii不动点定理.

Abstract:

In this paper,~we study the existence of positive solutions for a class of second-order nonlinear Neumann problem \[ \begin{cases} y''+a(t)y=\lambda g(t)f(y),~~t\in[0,1],\y'(0)=y'(1)=0, \end{cases} \] where~$\lambda>0$~is a positive parameter,~$f$~is superlinear at infinity,allowed to change sign， and the Green's function associated with this problem may vanish at some points. The proof of the main result is based on the Krasnosel'skii fixed-point theorem.