On the number of spikes concentrating on hyperplanes or line-segments to a Lin-Ni-Takagi problem

Abstract

We consider the following singularly perturbed elliptic Neumann problemwhere is the Laplace operator, is a constant, is the unit ball centered at the origin in with its unite outer normal , and f is superlinear and subcritical. For any integer m with , we show that when is sufficiently small, there exists a solution with k interior spikes located on , where k is bounded by with a positive constant depending only on N and f. In particular, when and , there exists a solution with at least interior spikes located on a hyperplane, which improves the result of Wang Y. in [Comm. Pure Appl. Anal. 2011;10:731–744]; when and , there exists a solution with at least interior spikes located on a line-segment, which improves the result of Ao W., Musso M. and Wei J. C. in [J. Differ. Eqs. 2011;251:881–901].

Keywords:

• multiple interior spikes
• singular perturbation
• localized energy method
• improved bound
• secondary Lyapunov-Schmidt reduction

AMS Subject Classifications:

• 35B40
• 35B25
• 35J25

Acknowledgements

 

Notes

This research is supported by National Natural Science Foundation of China [grant number 11171214] and the foundation of Nanjing Agricultural University [grant number LXYQ201300106].