# American Institute of Mathematical Sciences

September  2014, 34(9): 3371-3382. doi: 10.3934/dcds.2014.34.3371

## A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity

 1 Institut de Mathématiques de Toulouse & TSE, Université Toulouse I Capitole, Manufacture des Tabacs, 21, Allée de Brienne, 31015 Toulouse Cedex 6, France 2 Instituto de Matemática Interdiciplinar, Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040 Madrid, Spain

Received  December 2012 Revised  November 2013 Published  March 2014

We prove the compactness of the support of the solution of some stationary Schrödinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature.
Citation: Pascal Bégout, Jesús Ildefonso Díaz. A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3371-3382. doi: 10.3934/dcds.2014.34.3371
##### References:
 [1] S. N. Antontsev, J. I. Díaz, and S. Shmarev, Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations and their Applications, 48. Birkhäuser Boston Inc., Boston, MA, 2002. [2] P. Bégout and J. I. Díaz, Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity, Accepted for publication in RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., & arXiv:1304.3389. [3] P. Bégout and J. I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, Submitted, & arXiv:1301.0715. [4] P. Bégout and J. I. Díaz, On a nonlinear Schrödinger equation with a localizing effect, C. R. Math. Acad. Sci. Paris, 342 (2006), 459-463. doi: 10.1016/j.crma.2006.01.027. [5] P. Bégout and J. I. Díaz, Localizing estimates of the support of solutions of some nonlinear Schrödinger equations - The stationary case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 35-58. doi: 10.1016/j.anihpc.2011.09.001. [6] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.

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##### References:
 [1] S. N. Antontsev, J. I. Díaz, and S. Shmarev, Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations and their Applications, 48. Birkhäuser Boston Inc., Boston, MA, 2002. [2] P. Bégout and J. I. Díaz, Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity, Accepted for publication in RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., & arXiv:1304.3389. [3] P. Bégout and J. I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, Submitted, & arXiv:1301.0715. [4] P. Bégout and J. I. Díaz, On a nonlinear Schrödinger equation with a localizing effect, C. R. Math. Acad. Sci. Paris, 342 (2006), 459-463. doi: 10.1016/j.crma.2006.01.027. [5] P. Bégout and J. I. Díaz, Localizing estimates of the support of solutions of some nonlinear Schrödinger equations - The stationary case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 35-58. doi: 10.1016/j.anihpc.2011.09.001. [6] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
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