A multiplicity result for the scalar field equation

Advances in Nonlinear Analysis

article

Kanishka Perera¹ 

[1] Department of Mathematical Sciences, Florida Institute of Technology

关键词: Scalar field equation;    multiple nontrivial solutions;    variational and minimax methods;    concentration compactness;    symmetry breaking;   

DOI  :  10.1515/anona-2014-0022

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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 ≥ 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.

【 授权许可】

CC BY