Existence of nontrivial solutions for asymptotically linear periodic Schrödinger equations

Abstract

We study the Schrödinger equation:Section.Display where is periodic and is periodic in the -variables, is in a gap of the spectrum of the operator and is asymptotically linear as We prove that under some asymptotically linear assumptions for , this equation has a nontrivial solution. Our assumptions for are different from the classical assumptions raised by Li and Szulkin.

Keywords:

• semilinear Schrödinger equations
• linking
• asymptotically linear

AMS Subject Classifications:

• 35J20
• 35J60

Acknowledgments

The authors would like to thank the anonymous referees for their comments and suggestions on the manuscript. Shaowei Chen was supported by Science Foundation of Huaqiao University and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University.