Existence and multiplicity of solutions for fourth-order elliptic Kirchhoff equations with potential term

Abstract

In this paper, we consider the existence of nodal and multiple solutions for non-linear fourth-order elliptic Kirchhoff equationwhere . Making use of Mountain Pass Theorem, we establish existence of a non-trivial solution when is superlinear and subcritical. We also prove the existence of a positive solution, a negative solution and a nodal solution by using invariant sets of descending flow. Moreover, we show this nodal solution has a least energy and precisely two nodal domains. Under additional assumptions on , we obtain three existence results of infinitely many non-trivial and radial solutions via Fountain Theorem and Dual Fountain Theorem. We also show the existence of infinitely many nodal solutions by means of a sign-changing critical theorem.

Keywords:

• fourth-order Kirchhoff equation
• mountain pass theorem
• invariant sets of descending flow
• Fountain theorem
• dual Fountain theorem
• radial solution
• nodal solution

AMS Subject Classifications:

• 35J20
• 35J25
• 35J60
• 47J30

Acknowledgements

The authors would like to thank the referees for their useful and interesting comments and suggestions.