Multiple solutions of a Generalized singular perturbed Bratu Problem

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu′′+2u′+f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slow manifold theory, we reconsider the BVP within the framework of small and large values of the parameter. In the case of a special nonlinearity, numerical bifurcation patterns are studied that improve our understanding of the multivaluedness of the solutions.