Existence Theorem for a Class of Nonlinear Fourth-order Schrödinger-Kirchhoff-Type Equations.

Abstract

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type

\begin{equation*}\begin{cases}\Delta^{2}u-\left(a+b\displaystyle\int_{{\mathbb{R}}^N }|\nabla{u}|^2{\rm d}x\right)\Delta{u}+\lambda V(x)u=f(x,u),\quad x\in\mathbb{R}^N ,\\u\in{H^2({\mathbb{R}}^N)},\end{cases}\end{equation*}

where $a,b$ are positive constants, $\lambda \geq 1$ is a parameter, and the nonlinearity $f$ is either superlinear or sublinear  at infinity in $u$. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.