2013/11/14 by Kanishka Perera, Perera, Kanishka
Mathematics · #35P30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35J61 #Secondary 35J20 #math.AP #msc:35J20 #msc:35J61 #msc:35P30
paper · pdf · doi:10.48550/arxiv.1311.3587
arxiv created 2013/12/20 · arxiv updated 2013/12/23
We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in \mathbb 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.