Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space \begin{document}$ \mathbb{R}^N $\end{document} . The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by \begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document} where \begin{document}$ \alpha > 0, \beta \in \mathbb{R} $\end{document} and \begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document} is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where \begin{document}$ \beta $\end{document} is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.