Nonlinear elliptic problems on singularly perturbed domains

We consider how the shape of a domain affects the number of positive solutions of a nonlinear elliptic problem. In fact, we show that if a bounded domain Omega is sufficiently close to a union of disjoint bounded domains Omega (1),...,Omega (m), the number of positive solutions of a nonlinear elliptic problem on Omega is at least 2(m) - 1.