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Abstract
In this paper, we concern with the following quasilinear problemwhere
is the
-Laplacian of
for
,
is a small parameter,
is a continuous superlinear and subcritical nonlinearity. Suppose that
has at least one minimum and
has at least one maximum. We first prove that there are two families of positive solutions for
small, which concentrate, respectively, on the set of minimal points of
and the set of maximal points of
. In addition, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.
Acknowledgements
The authors would like to thank the referees for giving valuable comments and suggestions.
Notes
This work was supported by Natural Science Foundation of China [grant number 11201186]; Natural Science Foundation of Jiangsu Province [grant number BK2012282]; Jiangsu University foundation [grant number 11JDG117]; China Postdoctoral Science Foundation funded project [grant number 2012M511199], [grant number 2013T60499]; NSFC [grant number 11071038], [grant number 11171135].