where R is an open bounded subset of Rd, .Y is a linear real-valued uniformly elliptic differential operator, %9 is a linear differential boundary operator and the pair (Y’, 9) is formally self-adjoint. It is well-known that given L t R and .1‘ cC(R, R), (*) may have a nonunique solution (or may not have any solution at all). Recently some authors have studied the problem of finding the exact number of solutions of (*), for some special classes of (3’. 99) and ,f’ (see. e.g., Ambrosetti and Prodi [l], Amann 121, Ambrosetti and Mancini 131, Berger [4], Berestycki [s] and the references therein). The main purpose of the present paper is to continue this study (however, in the last two sections we turn to somewhat different problems). Let A, and iz be the first and the second eigenvalue of (Y,:#I), with ,I, simple. Suppose that J(t) is convex for t > 0, concave for 1 < 0 and .f‘(O) = ,f”(O) = 0. Furthermore. let