Ground state solutions for asymptotically periodic nonlinearities for Kirchhoff problems

Abstract

<p style='text-indent:20px;'>It is establish existence of ground state solutions for nonlocal elliptic problems driven by Kirchhoff problem in the following form:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u +V(x)u = \lambda q(x)u + g(x, u), \; \; \; x \in \mathbb{R}^3, u \in H^{1}(\mathbb{R}^{3}). \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the potential <inline-formula><tex-math id="M1">\begin{document}$ V $\end{document}</tex-math></inline-formula> and nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ g $\end{document}</tex-math></inline-formula> are periodic or asymptotically periodic. The main difficulty is to handle the lack of compactness due to the invariance under translations. The approach is based on minimization arguments over the Nehari set taking into account the fibering maps. Furthermore, due to the lack of compactness of Sobolev embedding into Lebesgue spaces, we need to recovery some kind of compactness required in variational methods. In order to do that we apply some fine estimates together with Lions' Lemma.</p>