We study the positive solutions to boundary value problems of the form -Δu=λf(u); Ω, α(x,u)(∂u/∂η)+[1-α(x,u)]u=0; ∂Ω, where Ω is a bounded domain in ℝn with n≥1, Δ is the Laplace operator, λ is a positive parameter, f:[0,∞)→(0,∞) is a continuous function which is sublinear at ∞, ∂u/∂η is the outward normal derivative, and α(x,u):Ω×ℝ→[0,1] is a smooth function nondecreasing in u. In particular, we discuss the existence of at least two positive radial solutions for λ≫1 when Ω is an annulus in ℝn. Further, we discuss the existence of a double S-shaped bifurcation curve when n=1, Ω=(0,1), and f(s)=eβs/(β+s) with β≫1.