The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.

Keywords: Nonlinear Schrödinger equation, Compact support, Energy method

@article{AIHPC_2012__29_1_35_0, author = {B\'egout, Pascal and D{\'\i}az, Jes\'us Ildefonso}, title = {Localizing estimates of the support of solutions of some nonlinear {Schr\"odinger} equations {\textendash} {The} stationary case}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {35--58}, publisher = {Elsevier}, volume = {29}, number = {1}, year = {2012}, doi = {10.1016/j.anihpc.2011.09.001}, mrnumber = {2876246}, zbl = {1241.35185}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.001/} }

TY - JOUR AU - Bégout, Pascal AU - Díaz, Jesús Ildefonso TI - Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 35 EP - 58 VL - 29 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.001/ DO - 10.1016/j.anihpc.2011.09.001 LA - en ID - AIHPC_2012__29_1_35_0 ER -

%0 Journal Article %A Bégout, Pascal %A Díaz, Jesús Ildefonso %T Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 35-58 %V 29 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.001/ %R 10.1016/j.anihpc.2011.09.001 %G en %F AIHPC_2012__29_1_35_0

Bégout, Pascal; Díaz, Jesús Ildefonso. Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 1, pp. 35-58. doi : 10.1016/j.anihpc.2011.09.001. http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.001/

[1] Energy methods for free boundary problems, Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications vol. 48, Birkhäuser Boston Inc., Boston, MA (2002) | MR | Zbl

, , ,[2] P. Bégout, J.I. Díaz, Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – the evolution case, in preparation.

[3] P. Bégout, J.I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, in preparation.

[4] On a nonlinear Schrödinger equation with a localizing effect, C. R. Math. Acad. Sci. Paris 342 no. 7 (2006), 459-463 | MR | Zbl

, ,[5] P. Bégout, V. Torri, Numerical computations of the support of solutions of some localizing stationary nonlinear Schrödinger equations, in preparation.

[6] Varias cuestiones sobre la ecuación de Schrödinger no lineal con coeficientes dependientes del espacio, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{\mathrm{e}}$MA 52 (2010), 97-128

,[7] Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 no. 2 (1979), 137-151 | MR | Zbl

, ,[8] Finite time extinction by nonlinear damping for the Schrödinger equation, Comm. Partial Differential Equations 36 no. 6 (2011), 961-975 | MR | Zbl

, ,[9] Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University Courant Institute of Mathematical Sciences, New York (2003) | MR | Zbl

,[10] An Introduction to Semilinear Elliptic Equations, Editora do Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro (2006)

,[11] Nonlinear Partial Differential Equations and Free Boundaries, vol. I, Elliptic Equations, Research Notes in Mathematics vol. 106, Pitman (Advanced Publishing Program), Boston, MA (1985) | MR | Zbl

,[12] Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc. 290 no. 2 (1985), 787-814 | MR | Zbl

, ,[13] Convex Analysis and Variational Problems, Classics in Applied Mathematics vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999) | MR | Zbl

, ,[14] Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin (2001) | MR | Zbl

, ,[15] Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics vol. 24, Pitman (Advanced Publishing Program), Boston, MA (1985) | MR | Zbl

,[16] Definitions of maximal differential operators, Ark. Mat. 3 (1958), 501-504 | MR | Zbl

,[17] Propagation estimates for Schrödinger-type operators, Trans. Amer. Math. Soc. 291 no. 1 (1985), 129-144 | MR | Zbl

,[18] Compactification of nonlinear patterns and waves, Phys. Rev. Lett. 101 no. 26 (2008), 261602

, ,[19] Dissipation at singularities of the nonlinear Schrödinger equation through limits of regularisations, Phys. D 138 no. 3–4 (2000), 334-343 | MR | Zbl

,[20] Problèmes aux limites non homogènes, II, Ann. Inst. Fourier (Grenoble) 11 (1961), 137-178 | EuDML | MR | Zbl

, ,[21] Problemi ai limiti non omogenei, III, Ann. Sc. Norm. Super. Pisa (3) 15 (1961), 41-103 | EuDML | Numdam | MR | Zbl

, ,[22] Schrödinger operators with singular complex potentials as generators: existence and stability, Semigroup Forum 60 no. 3 (2000), 337-343 | MR | Zbl

, ,[23] Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit, Chaos 15 no. 1 (2005), 015111 | MR | Zbl

, ,[24] Partial Differential Equations. An Introduction, John Wiley & Sons Inc., New York (1992) | MR

,[25] The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences vol. 139, Springer-Verlag, New York (1999) | MR | Zbl

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