On s-shaped bifurcation curves for multiparameter positone problems

摘要

We study the existence of multiple positive solutions to the two point boundary value problem -u″(x) = ⋋f(u(x)); O< x < 1u(0) = 0 = u(1) + αu′(1), where ⋋ > 0, α > 0. Here f is a smooth function such that f > 0 on [0, r) for some 0 < r ≤ ∞. In particular, we consider the case when f is initially convex and then concave. We discuss sufficient conditions for the existence of at least three positive solutions for a certain range (independent of α) of λ. We apply our results to the nonlinearity f(u) =exp[cu(c+u)]; c > 4 which arises in combustion theory and to the nonlinearity f(u) = (σ−u)exp{-c(1+u)}; σ > 0 (fixed), c >4+4σ, which arises in chemical reactor theory.