A Class Of Exact Number Of Single-parameter Dirichlet Boundary Value Problem Solution

By use of the time–mapping method, we study the complete classification of theexact number of positive solutions for the boundary value problemwhere p > 1, andλ≥0 is a parameter. We assume that the function f satisfies thefollowing conditions:(H0) f : [0,∞)×[0,∞)â†'(?∞,+∞) is continuous. For any fixedλ≥0, f(λ,u)has continuous second order partial derivatives in (0,∞) with respect to u and(H1) For any fixedλ≥0, there exists V0(λ) > 0 such thatand(H2) For any fixed u > 0, f(λ,u) is strictly increasing in [0,∞) with respect toλand(H3) (p - 1)f(0,u) - uf_u(0,u) < 0 for u > 0.(H4) For any fixedλ> 0, one of the following two conditions is satisfied.(H4a) There exists U0(λ) > 0 such thatand is nondecreasing in (U0(λ),∞) with respect to u.(H4b) (H5) For any fixedλ≥0,(H6) The functionis continuous and nonincreasing on [0,∞) with respect toλ.