Nonlinear scalar field equations with competing potentials

In this talk a class of nonlinear elliptic equations on R^N is presented. These equations come from physical problems, where a potential interacts with the bumps in the solutions. We first discuss the interactions and present some old and new results; then we consider competing potentials, showing in particular a theorem on the existence of infinitely many positive bound state solutions. Finally, we present some examples on existence/non existence of a ground state solution, when the theorem applies.