Threshold scattering for the 2d radial cubic-quintic NLS

We consider the cubic-quintic nonlinear Schr\"odinger equation in two space dimensions. For this model, X. Cheng established scattering for $H^1$ data with mass strictly below that of the ground state for the cubic NLS. Subsequently, R. Carles and C. Sparber utilized the pseudoconformal energy estimate to obtain scattering at the sharp threshold for data belonging to a weighted Sobolev space. In this work, we remove the weighted assumption and establish scattering at the threshold for radial data in $H^1$.Comment: 19 page

Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions

AbstractIn [T. Duyckaerts, F. Merle, Dynamic of threshold solutions for energy-critical NLS, preprint, arXiv:0710.5915 [math.AP]], T. Duyckaerts and F. Merle studied the variational structure near the ground state solution W of the energy critical NLS and classified the solutions with the threshold energy E(W) in dimensions d=3,4,5 under the radial assumption. In this paper, we extend the results to all dimensions d⩾6. The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W

Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow up elements at a non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$