Existence of positive solutions for singular higher-order differential equations

In this paper, we consider the existence of positive solutions for the following singular higher-order eigenvalue problem (HEP): {(−1)nu(2n)(t)=λa(t)f(t,u(t),u″(t),…,u(2(n−1))(t)),00λ>0 is a parameter, a(t)a(t) may have singularity at t=0t=0 and or 1; furthermore, f(t,v1,v2,…,vn)f(t,v1,v2,…,vn) may also be singular at vi=0(i=1,2,…,n). Without making any monotone-type assumptions, we establish various lemmas and theorems for the existence of positive solutions of the HEP when λλ is in an explicit interval. In addition, the associated Green function for the above problem is derived.