In this chapter we present applications of the Leray-Schauder type theorems to the weak solvability of the semilinear Dirichlet problem

under the assumption that the constant c is not an eigenvalue of -A (nonresonance condition) and that the growth of f (X, U , v) on U and v is at most linear. Here A is the Laplacian

and V is the gradient

Such results have been obtained by many authors, see [25], [29], [70], [98] for example. We particularly refer the reader to the papers [62], [91] and [137].