Existence and multiplicity results for Kirchhoff type problems with four-superlinear potentials

Abstract

In this article, we study the existence and multiplicity results for Kirchhoff type equations of the form

where is three-superlinear at infinity. Under certain assumptions on f, we obtain at least one nontrivial solution for (*) without assuming the Ambrosetti and Rabinowitz condition by using the mountain pass theorem, and obtain infinitely many solutions when f(x, ·) is odd by using the fountain theorem.

Keywords:

• Kirchhoff type equation
• mountain pass theorem
• fountain theorem
• variational methods

AMS Subject Classifications:

• 35J65
• 35J20
• 47J10

Acknowledgement

The authors thank the referees for careful reading of the manuscript and for valuable suggestions. The research was supported by NSFC (11171047), NSF of Education Committee of Henan province (12B11026) and NSF of Henan province (122300410034).