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

Boundary Value Problems

Ru Yuanfang¹  An Yukun² 

[1] College of Science, China Pharmaceutical University;College of Science, Nanjing University of Aeronautics and Astronautics;

关键词: Fourth-order elliptic equation;    Nonlocal;    Asymptotically linear;    Mountain pass theorem;    Critical point;   

DOI  :  10.1186/s13661-020-01430-4

来源: DOAJ

【 摘 要 】

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

【 授权许可】

Unknown