奇异超线性和次线性n阶m点边值问题的非平凡解

Nontrivial Solutions of Singular Superlinear and Sublinear n-order m-point Boundary Value Problems

在相应线性算子第一特征值的条件下,讨论超线性和次线性n阶m点边值问题{u(n)(t)+a(t)f(u(t))=0,t∈(0,1)m-2,其中:n≥2,m≥2,0<η1<η2<…<u(0)=u'(0)=…=u(n-2)(0),u(1)=∑αiu(ηi)i=1m-2ηm-2<1,αi>0,(i=1,2,…,m-2)且∑αiηn-1i<1.在此允许a(x)在x=0和x=1奇异,f不i=1必是非负的.利用锥上的拓扑度理论获得非平凡解的存在性.

In this paper, the singular supedinear and sublinear n-order m-point boundary value problem （{u（n）（t）＋a（t）f（u（t））=0,t∈（0,1） u（0）=u＇（0）=…=u（n-2）（0）,u（1）=m-2∑i=1αiu（ηi）） is considered under some conditions concern n≥2,m≥2,0〈η1〈η2〈…〈ηm-2〈1,αi〉0,（i=1,2,…,m-2） and m-2∑αiηn-1i〈1 is allowed to be singular at x = 0 and x = 1 and f（x） is not necessary to be nonnegative. The existence results of nontrivial solutions and positive solutions are given by using the method of topological degree.