On the least energy solutions for semilinear Schrödinger equation with electromagnetic fields involving critical growth and indefinite potentials

Abstract

In this paper, we are concerned with the following semilinear Schrödinger equation with electromagnetic fields and critical growth

for sufficiently large , where , and its zero set is not empty, is the critical Sobolev exponent, is a constant such that the operator might be indefinite but is non-degenerate. Using variational method and modified Nehari–Pankov method, we prove the equation admits a least energy solution which localizes near the potential well . The results we obtain here extend the corresponding results for the Schrödinger equation which involves critical growth but does not involve electromagnetic fields.

Keywords:

• Semilinear Schrödinger equation
• least energy solution
• potential well
• critical growth
• electromagnetic fields

AMS Subject Classifications:

• 35J60
• 35B33

Acknowledgements

The authors first want to express their gratitude to the referee and the editor for their valuable comments and suggestions. On the other hand, this paper was finished while the first author visited School of Mathematical Sciences of Beijing Normal University as a visiting fellow. The first author would like to express her gratitude for their hospitality during her visit.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Science and Technology Planning Project of Gansu Province [grant number 1610RJZA102]; Fundamental Research Funds for the Central Universities [grant number 31920170147]; National Science Foundation of China [grant number 11571040].