Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in $\mathbb{R}^{N}$

Abstract

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N}, \end{eqnarray*} where $M(t):\mathbb{R}\rightarrow\mathbb{R}$ is the Kirchhoff function, $f(x,u)=\lambda k(x,u)+ h(x,u)$, $\lambda\geq0$, $k(x,u)$ is of sublinear growth and $h(x,u)$ satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for $\lambda=0$. For $\lambda>0$ small enough, we obtain at least two nontrivial solutions. Furthermore, if $f(x,u)$ is odd in $u$, we show that above equations possess infinitely many solutions for all $\lambda\geq0$. Our theorems generalize some known results in the literatures even for $\lambda=0$ and our proof is based on the variational methods.