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

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- by Pierre Raphaël and Jeremie Szeftel
- J. Amer. Math. Soc.
**24**(2011), 471-546 - DOI: https://doi.org/10.1090/S0894-0347-2010-00688-1
- Published electronically: December 2, 2010
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## Abstract:

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 a 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 nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case $k\equiv 1$.## References

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## Bibliographic Information

**Pierre Raphaël**- Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
- Email: pierre.raphael@math.univ-toulouse.fr
**Jeremie Szeftel**- Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75 005 Paris, France
- MR Author ID: 712495
- Email: jeremie.szeftel@ens.fr
- Received by editor(s): January 4, 2010
- Received by editor(s) in revised form: September 2, 2010
- Published electronically: December 2, 2010
- Additional Notes: The first author was supported by the French Agence Nationale de la Recherche, ANR jeunes chercheurs SWAP and by ANR OndeNonLin

The second author was supported by the French Agence Nationale de la Recherche, ANR jeunes chercheurs SWAP - © Copyright 2010 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**24**(2011), 471-546 - MSC (2010): Primary 35B35, 35B44; Secondary 35Q41, 35Q55
- DOI: https://doi.org/10.1090/S0894-0347-2010-00688-1
- MathSciNet review: 2748399