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.