ON S-SHAPED BIFURCATION CURVES FOR MULTIPARAMETER POSITONE PROBLEMS

We study the existence of multiple positive solutions to the two point boundary value problem -u''(x) = lambda f(u(x)); 0 < x < 1 u(0) = 0 = u(1) + alpha u'(1) where lambda > O, alpha > O. Here f is a smooth fu nction such that f > 0 on [O, r) for some 0 < r less than or equal to infinity. In particular, we consider the case when f is initially conv ex and then concave. We discuss sufficient conditions for the existenc e of at least three positive solutions for a certain range (independen t of ct) of X. We apply our results to the nonlinearity f(u) = exp[cu/ (c+ u)]; c > 4 which arises in combustion theory and to the nonlineari ty f(u) = (sigma - u) exp{-c/(1 + u)}; sigma > O (fixed), c > 4 + 4/si gma, which arises in chemical reactor theory.