Existence and multiplicity results for a class of quasilinear Schrödinger equations in involving critical growth

Abstract

The existence of multi-bump solutions for the following class of quasilinear Schrödinger equations in is established, where and h is a continuous function, are continuous functions verifying some hypotheses. By a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that if the zero set of V(x) has several isolated connected components such that the interior of is not empty and is smooth, then for large there exists, for any non-empty subset , a bump solution which is trapped in a neighbourhood of .

Keywords:

• Quasilinear Schrödinger equation
• critical growth
• multi-bump solutions
• variational methods

AMS Subject Classifications:

• 35J20
• 35J60
• 35Q55

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China [grant number 11301038]; The Natural Science Foundation of Jilin Province [grant number 20160101244JC]; The second author is supported by NSFC [grant number 11371166].