Calc. Var. 2, 29--48 (1994) @ Springer-Verlag 1994 Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology Vieri Benci I and Giovanna Cerami 2 1 Universit~t di Pisa, Istituto di Mathematiche Appliccata, Facolt~ di Ingegneria, Via Bonanno 25/B, 1-56100 Pisa, Italy 2 Universit~ di Palermo, Facolt~ die Ingegneria, Viale delle Scienze, 1-90100 Palermo, Italy Received April 19, 1993/Accepted April 29, 1993 Abstract. We use Morse theory to estimate the number of positive solutions of an elliptic problem in an open bounded set f2 C JR N. The number of solutions depends on the topology of f?, actually on ~t(f2), the Poincar6 polynomial of X-2. More precisely, we obtain the following Morse relations: Z t/z(u) : ~t(~) + ~2 [~(~) __ 1] + t(1 + t) ~(t), uE,~C where ~(t) is a polynomial with non-negative integer coefficients, ~ is the set of positive solutions of our problem and #(u) is the Morse index of the solution u. Mathematics Subject Classification." 35J20, 49F15, 58E05 1 Introduction In this paper we are concerned with the following problem: eA+u=f(u) in~, (P~) u > 0 in X? u = 0 on 0~2 where c 6 IR+\{0}, f2 C IR N, N _> 3, is a smooth bounded domain and f:R + ,.~ is a ~l'l-function with f(O) = f'(O) = O. Precisely the present research continues a study of [B.C.], [B.C.P.], and [C.P.] on the effect of the domain shape on the number of positive solutions of some semilinear elliptic problems. During the past few years the relations between the geometry or the topology of f2 and the existence and multiplicity of solutions to problems like the following