Multiple solutions to the Bahri-Coron problem in some domains with nontrivial topology

We show that in every dimension $N\geq3$ there are many bounded domains $\Omega\subset\mathbb{R}^{N}$, having only finite symmetries, in which the Bahri-Coron problem \[-\Delta u=|u| ^{4/(N-2)}u\text{\in}\Omega,\text{\ \}u=0\text{\ on}\partial\Omega, \] has a prescribed number of solutions, one of them being positive and the rest sign changing.