Global well-posedness, scattering and blow-up for the energy-critical, Schr\"odinger equation with indefinite potential in the radial case

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u +|u|^{\frac{4}{N-2}}u=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. Moreover, we get a positive ground state solution which is radially symmetric by using variational methods. This paper extends the results of \cite{KCEMF2006}(Invent. Math) to the potential equation and develops the recent conclusions.