Infinitely many high energy solutions for nonlocal fourth-order equation with sigh-changing potential

Abstract

This paper is devoted to a class of general nonlocal fourth-order elliptic equation ${\mathrm{\Delta }}^{2}u-(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{\Delta }u+V(x)u=f(x,u),x\in {\mathbb{R}}^N,$where ${\mathrm{\Delta }}^2=\mathrm{\Delta }(\mathrm{\Delta })$ is the biharmonic operator, $a>0,b\ge 0$ are constants and the potential V is allowed to be sign-changing. Under certain assumptions on f, infinitely many high energy solutions are established. Our results enrich that obtained in the literature.

Keywords:

• Biharmonic operator
• sign-changing potential
• symmetric mountain pass theorem
• infinitely many solutions

2010 Mathematics Subject Classifications:

• 35B38
• 35A15
• 35J60

Disclosure statement

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

Additional information

Funding

The research was supported by Youth Natural Science Foundation of Shanxi Province (No. 20210302124527), the Science and Technology Innovation Project of Shanxi (No. 2020L0260) and Youth Science Foundation of Shanxi University of Finance and Economics (No. QN-202020).