On nonlinear perturbations of a periodic integrodifferential equation with critical exponential growth

Abstract

In this paper, we study the existence of solutions for integrodifferential Schrödinger equations of the form $-{\mathcal{L}}_{K}u+V\left(x\right)u=f(x,u)\text{in}\mathbb{R},$where $-{\mathcal{L}}_K$ is a nonlocal operator with a measurable kernel which satisfies ‘structural properties’, more general than the standard kernel of fractional Laplacian operator, V is a bounded potential, and the nonlinear term $f(x,u)$ has critical exponential growth with respect to the Trudinger–Moser inequality.

Keywords:

• Integrodifferential operators
• variational methods
• critical points
• Trudinger–Moser inequality

2010 Mathematics Subject Classifications:

• 35J60
• 35R11
• 58E30

Acknowledgement

The authors would like to thank Professor Manassés Xavier de Souza for his most valuable comments and suggestions in the preparation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).