Nontrivial solutions for a fourth-order elliptic equation of Kirchhoff type via Galerkin method

In this paper we study the existence of nontrivial solutions of the nonlocal elliptic problem Δ2u-(a+b∫Ω|∇u|2dx)Δu+u=(∫Ωg(x,u)dx)γf(x,u),inΩ,u=Δu=0,on∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lcr} \Delta ^2 u-(a+b\int _\Omega |\nabla u|^2\mathrm{d}x)\Delta u+u=(\int _\Omega g(x,u)\mathrm{d}x)^\gamma f(x,u),\;in\;\Omega ,\\ u=\Delta u=0,\;\;on\;\;\partial \Omega \end{array}\right. \end{aligned}$$\end{document}via Galerkin method, where Ω⊂RN(N≥5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^N (N\ge 5)$$\end{document} is a smooth bounded domain.