奇异sturm-Liouville特征值问题正解的全局分歧和存在性

The Global Bifurcation and the Existence of Positive Solutions of Singular Sturm-Liouville Eigenvalue Problems

本文研究了奇异Sturm-Liouville特征值问题{u''(t)+λa(t)f(u(t))=0,0<t<1,u(0)-βu'(0)=0,u(1)+γu'(1)=αu(η),其中λ>0是参数,α,β,γ≥0,0≤η≤1;α∈C((0,1),(0,+∞))在t=0和/或t=1处可能有奇性,f∈C([0,+∞),(0,+∞)).文中首先给出了正解的一些精确的先验估计和渐近行为分析.再利用这些结果联合不动点指数定理证明了正解的全局存在性.一个关键的技术是利用连续统构造上下解.

In this paper, we study the singular Sturm-Liouville eigenvalue problems {u″＋λa（t）f（u（t））=0, 0〈t〈1, u（0）-βu′（0）=0, u（1）＋γu′（1）=au（η）, where λ〉0 is a parameter, α,β,γ≥0,0〈η〈1;α∈C（（0,1）,（0,＋∞）） may havesingular at t = 0 and/or t = 1, and f E C（[0, ＋∞）, （0, ＋∞））. We first give some sharp a priori estimates and asymptotical behavior of positive solutions. Those combining with the fixed point index theorem can investigate the global existence results of positive solutions. A key technique is the construction of the lower and upper solutions by the use of the continuum.