Dynamics for the focusing, energy-critical nonlinear Hartree equation

Abstract

In \cite{LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har}, the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when $E(u_0)<E(W)$, where $W$ is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy. Our arguments closely follow those in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution, DuyRouden:NLS:ThresholdSolution, LiZh:NLS, LiZh:NLW}. The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in $L^{\frac{2d}{d-2}}({\mathbb{R}}^d)$ is regular and unique by the moving plane method in its global form, which plays an important role in the spectral theory of the linearized operator and the dynamics behavior of the threshold solution.

53 pages

Cited by in corpus (3)

• The dynamics of the 3D radial NLS with the combined terms
• Dynamics of threshold solutions for energy critical NLS with inverse square potential
• Global well-posedness for the defocusing Hartree equation with radial data in ${\mathbb{R}}^4$