Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation

Abstract

Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrodinger equation (INLS) \begin{document}$ \begin{equation*} i\partial_{t}u+\Delta u+|x|^{-b}|u|^{p-1}u = 0. \end{equation*} $\end{document} We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in \begin{document}$ H^{1}(\mathbb{R^{N}}) $\end{document} in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is \begin{document}$ L^{2} $\end{document} -supercritical, then the ground states are strongly unstable by blow-up.