  We consider a weakly coupled singularly perturbed variational elliptic system
in a bounded smooth domain with Dirichlet boundary conditions. We show that, in
the competitive regime, the number of fully nontrivial solutions with
nonnegative components increases with the number of equations. Our proofs use a
combination of four key elements: a convenient variational approach, the
asymptotic behavior of solutions (concentration), the Lusternik-Schnirelman
theory, and new estimates on the category of suitable configuration spaces.
