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

Abstract

<p style='text-indent:20px;'>Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS) <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} i\partial_{t}u+\Delta u+|x|^{-b}|u|^{p-1}u = 0. \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in <inline-formula><tex-math id="M1">\begin{document}$ H^{1}(\mathbb{R^{N}}) $\end{document}</tex-math></inline-formula> in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is <inline-formula><tex-math id="M2">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-supercritical, then the ground states are strongly unstable by blow-up.