Existence and multiplicity of solutions for nonlocal fourth-order elliptic equations with combined nonlinearities

Abstract

AbstractThis paper is concerned with the following nonlocal fourth-order elliptic problem: $$\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u-m(\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\Delta u=a(x) \vert u \vert ^{s-2}u+f(x,u), \quad x\in \varOmega , \\ u=\Delta u=0,\quad x\in \partial \varOmega , \end{cases}\displaystyle \end{aligned}$$ { Δ 2 u − m ( ∫ Ω | ∇ u | 2 d x ) Δ u = a ( x ) | u | s − 2 u + f ( x , u ) , x ∈ Ω , u = Δ u = 0 , x ∈ ∂ Ω , by using the mountain pass theorem, the least action principle, and the Ekeland variational principle, the existence and multiplicity results are obtained.