Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Abstract

Using variational methods, we establish existence of multi-bump solutions for the following class of problems

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2 u +(\lambda V(x)+1)u = f(u), \quad \text{ in } \quad \mathbb {R}^{N},\\ u \in H^{2}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$

where $N \ge 1$, $\Delta ^2$ is the biharmonic operator, f is a continuous function with subcritical growth, $V : \mathbb {R}^N \rightarrow \mathbb {R}$ is a continuous function verifying some conditions and $\lambda >0$ is a real constant large enough.