A note on nonlinear fourth-order elliptic equations on $\mathbb R ^N$

Abstract

We established the existence of weak solutions of the fourth-order elliptic equation of the form

$$\begin{aligned} \Delta ^2 u -\Delta u + a(x)u = \lambda b(x) f(u) + \mu g (x, u), \qquad x \in \mathbb{R }^N, u \in H^2(\mathbb{R }^N), \end{aligned}$$

where $\lambda$ is a positive parameter, $a(x)$ and $b(x)$ are positive functions, while $f : \mathbb{R }\rightarrow \mathbb{R }$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that the problem has at least three solutions.