Existence of Nonnegative Solutions for Fourth Order Elliptic Equations of Kirchhoff Type with General Subcritical Growth

Abstract

This paper is dedicated to investigating the following fourth-order elliptic equation with Kirchhoff-type \[ \begin{cases} \displaystyle \Delta^{2} u - \left( a + b \int_{\mathbb{R}^{N}} |\nabla u|^{2} \, dx \right) \Delta u + cu = f(u) &\textrm{in $\mathbb{R}^{N}$}, \\ u \in H^{2}(\mathbb{R}^{N}), \end{cases} \] where $a \gt 0$, $b \geq 0$ and $c \gt 0$ are constants. By using cut-off functional and monotonicity tricks, we prove that the above problem has a positive solution. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear condition at the infinity, which extend the results of related literatures.