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 $$ \left\{ \begin{array}{l} \Delta^2 u +(\lambda V(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{N}, u \in H^{2}(\mathbb{R}^{N}), \end{array} \right. $$ where $N \geq 1$, $\Delta^2$ is the biharmonic operator, $f$ is a continuous function with subcritical growth and $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous function verifying some conditions.