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

Abstract

We use Morse theory to estimate the number of positive solutions of an elliptic problem in an open bounded setΩ ∉ ℝ^N. The number of solutions depends on the topology ofΩ, actually onP _t (Ω), the Poincaré polynomial ofΩ. More precisely, we obtain the following Morse relations:

$$\sum\limits_{u \in K} {t^{\mu \left( u \right)} } = tP_t \left( \Omega \right) + t^2 [P_t \left( \Omega \right) - 1] + t\left( {1 + t} \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \left( t \right)$$

, where$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \left( t \right)$ is a polynomial with non-negative integer coefficients,K is the set of positive solutions of our problem andμ(u) is the Morse index of the solutionu.