Critical nonhomogeneous fourth-order Schrödinger–Kirchhoff-type equations

In this paper we study the following class of stationary fourth-order Schrödinger–Kirchhoff-type equations: Δ2u-M‖∇u‖22Δu+V(x)u=h(x)|u|q-2u+|u|2∗-2u+g(x)|u|τ-2u,x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^{2} u-M\left( \Vert \nabla u\Vert ^2_2 \right) \Delta u+V(x)u=h(x)|u|^{q-2}u+|u|^{2_*-2}u+ g(x)|u|^{\tau -2}u, ~~x \in \mathbb {R}^{N}, \end{aligned}$$\end{document}where N≥8,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 8,$$\end{document} and 2∗=2NN-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2_*=\frac{2N}{N-4}$$\end{document} is the critical Sobolev exponent. Under some assumptions on the Kirchhoff function M, the potential V(x) and g(x), by using Ekeland’s Variational Principle and the Mountain Pass Theorem, we obtain the existence of multiple solutions for the above problem. These results are new even for the local case, which corresponds to nonlinear fourth order Schrödinger equations.