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

A multiplicity result for the scalar field equation