New results concerning a singular biharmonic equations with p-Laplacian and Hardy potential

Abstract

This paper deals with a singular biharmonic equations with p-Laplacian and Hardy potential. Using Mountain Pass Theorem and Fountain Theorem with Cerami condition, we obtain the existence and multiplicity of sign-changing high-energy solutions under some suitable conditions on the nonlinear term $f(x,u)$. In addition, by applying an abstract critical point theorem of Kajikiya in [A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370], a sequence of sign-changing solutions converging to zero for the biharmonic equations with sublinear nonlinearities is also obtained.

Keywords:

• Biharmonic operator
• mountain pass theorem
• fountain theorem
• sign-changing solution

Mathematics Subject Classifications:

• 35B38
• 35A15
• 35J60

Acknowledgments

The authors would like to thank the anonymous referee(s) for reading the manuscript carefully and giving some useful suggestions.

Disclosure statement

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

Additional information

Funding

The research was supported by Hunan Provincial Natural Science Foundation of China [grant number 2019JJ40068], Hunan Provincial Educational Foundation of China [grant number 21A0361].