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].
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].