Focusing nlkg equation with singular potential

Abstract

We study the dynamics for the focusing nonlinear Klein Gordon equation with a positive, singular, radial potential and initial data in energy space. More precisely, we deal with \begin{document}$u_{tt}-Δ u+m^2 u=|x|^{-a}|u|^{p-1}u$ \end{document} with \begin{document}$0 . In dimension \begin{document}$d≥3$\end{document} , we establish the existence and uniqueness of the ground state solution that enables us to define a threshold size for the initial data that separates global existence and blow-up. We find a critical exponent depending on \begin{document}$a$\end{document} . We establish a global existence result for subcritical exponents and subcritical energy data. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary sets.