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

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential equation* \array{l i ₜ u+ u-V (x) u +|u|^4{N-2}u=0, \ (x, t) RN R, \\. u|ₓ=₀=u₀ H ¹ (RN), array. equation* where V (x): RN 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 KCEMF2006 (Invent. Math) to the potential equation and develops the recent conclusions.