Existence of Solutions to u + u + g t u u = p t u 0 = u π = 0

Existence and multiplicity results for the boundary value problem u + u + g t u u = p t 0 < t < π u 0 = u π = 0 are presented. The proofs are based on the alternative method, a connectedness result, the contraction mapping principle, and a detailed analysis of the bifurcation equation utilizing, e.g., a generalization of the mean value theorem for integrals. We shall obtain results with g bounded or unbounded, having finite limits at ±∞ or without limits, thus extending some recent results in the literature. The proofs offer a constructive way to find the bounds for p¯ and to find numerically the number of solutions and the approximative solutions.