Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology

We use Morse theory to estimate the number of positive solutions of an elliptic problem in an open bounded set OMEGA subset-of R(N). The number of solutions depends on the topology of OMEGA, actually on P(t)(OMEGA), the Poincare polynomial of OMEGA. More precisely, we obtain the following Morse relations: SIGMA(u is-an-element-of K) t(mu(u)) = tP(t)(OMEGA) + t2 [P(t)(OMEGA) - 1] + t(1 + t) Q(t), where Q(t) is a polynomial with non-negative integer coefficients, K is the set of positive solutions of our problem and mu(u) is the Morse index of the solution u.