A multiplicity result for the scalar field equation

作者：Perera Kanishka^*

来源：Advances in Nonlinear Analysis, 2014, 3(S1): S47-S54.

DOI：10.1515/anona-2014-0022

• Scalar field equation
• multiple nontrivial solutions
• variational and minimax methods
• concentration compactness
• symmetry breaking

We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in 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 %26gt;= 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.

• 出版日期2014-9