A multiplicity result for the scalar field equation

• Main Author: Perera Kanishka
• Format: Article
• Language: English
• Published: De Gruyter 2014-09-01
• Series: Advances in Nonlinear Analysis
• Subjects: scalar field equation; multiple nontrivial solutions; variational and minimax methods; concentration compactness; symmetry breaking; 35j61; 35p30; 35j20
• Online Access: https://doi.org/10.1515/anona-2014-0022

We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in ℝN 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 ≥ 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.