A multiplicity result for the scalar field equation

We prove the existence of $N - 1$ distinct pairs of nontrivial solutions of the scalar field equation in ${\mathbb R}^N$ under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when $N \ge 6$. When the ground state is the only positive solution, we also obtain the stronger result that at least $N - 1$ of the first $N$ minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences.