Global structure of solutions for a class of two-point boundary value problems involving singular and convex or concave nonlinearities

Abstract

This paper investigates the following two-point singular boundary value problems (BVP): { − x ″ ( t ) = λ x q + x p , t ∈ ( 0 , 1 ) , x ( t ) > 0 , t ∈ ( 0 , 1 ) , x ( 0 ) = x ( 1 ) = 0 , where q < 0 and p > 0 are fixed given numbers; λ ∈ R + = [ 0 , + ∞ ) is a parameter. The results obtained are the global structure of solutions and exact number of solutions when p ⩽ 1 or p > 1 and q is sufficiently small.