On the relation between interior critical points and parameters for a class of nonlinear problems with Neumann-Robin boundary conditions

We consider boundary value problem-u″(x)=λf(u(x)),x∈(0,1),u′(0)=0,u′(1)+αu(1)=0,where α ⩾ 0, λ > 0 are parameters and f ∈ C2[0, ∞) such that f(0) < 0. In this paper we study for the cases p ∈ (0, β) and p ∈ (β, θ) (p is the value of the solution at x = 0 and β, θ are such that f(β) = 0, F(θ)=∫0θf(t)dt=0), the relation between λ and the number of interior critical points of the positive solutions of the above system.