  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.
