The author proves some existence theorems concerning the nonlinear Dirichlet boundary value problem
and the unbounded nonlinearity f is a Carathéodory function for
satisfying certain growth conditions. The main tool is a coincidence degree result due to J. Mawhin
[Topological degree methods in nonlinear BVPs, CBMS regional Conf. No.40, Am. Math. Soc. (1970)].