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.