Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-09T22:02:24.524Z Has data issue: false hasContentIssue false
  • English
  • Français

On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation*

Published online by Cambridge University Press:  14 November 2011

Russell Johnson  and
Xingbin Pan
Russell Johnson
Affiliation:
Dipartimento di Sistemi e Informatica, Universita di Firenze, Via di Santa Marta 3, 50139 Firenze, Italy
Xingbin Pan
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, People's Republic of China
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in ℝn. The equation is derived from the blow-up problem in the non-linear Schrödinger equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 4 , 1993 , pp. 763 - 782
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Akhmanov, S. A., Sukharukov, A. P. and Kokhlov, R. V.. Self-focusing and self-trapping of intense light beams in a nonlinear medium. Translated in Soviet Phys. JETP 23 (1966), 1025–1033.Google Scholar
2Chiao, R. Y., Garmire, E. and Townes, C. H.. Self-trapping of optical beams. Phys. Rev. Lett. 13 (1964), 479–482.CrossRefGoogle Scholar
3Elliot, C. J. and Suydam, B. R.. Self-focusing phenomena in air-glass laser structures. IEEE J. Quantum Electronics 11 (1975), 863–866.CrossRefGoogle Scholar
4Glassey, R. T.. On the blowing up of the solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18 (1977), 1974–1977.CrossRefGoogle Scholar
5Gidas, B., Ni, W.-M. and Nirenberg, L.. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Adv. Math. Stud. 7A (1981), 369–402.Google Scholar
6Ginibre, J. and Velo, G.. On a class of nonlinear Schrödinger equations, I. The Cauchy problem, general case. J. Fund. Anal. 32 (1979), 1–32.CrossRefGoogle Scholar
7Johnson, R. A., Pan, X.-B. and Yi, Y.-F.. Singular solutions of the elliptic equation ∆u − u + u p = 0. Georgia Inst. Tech. Math. Report Ser. CDSNS 91–45.Google Scholar
8Kwong, M.-K.. Uniqueness of positive solutions of ∆u − u + u p = 0 in Rp. Arch. Rational Mech. Anal. 105 (1989), 243–266.CrossRefGoogle Scholar
9Levine, H.A.. The role of critical exponents in blow up theorems. Siam Rev. 32 (1990), 262–288.CrossRefGoogle Scholar
10LeMesurier, B., Papanicolaou, G., Sulem, C. and Sulem, P.-L.. The focusing singularity of the nonlinear Schrödinger equation. In Directions in Partial Differential Equations, eds Crandall, M. G.et al, 159–201 (New York: Academic Press 1981).Google Scholar
11LeMesurier, B., Papanicolaou, G., Sulem, C. and Sulem, P. L.. Focusing and multi-focusing solutions of the nonlinear Schrödinger equation. Phys. D 31 (1988), 78–102.CrossRefGoogle Scholar
12LeMesurier, B., Papanicolaou, G., Sulem, C. and Sulem, P.-L.. Local structure of the self-focusing singularity of the nonlinear Schrödinger equation. Phys. D 32 (1988), 210–226.CrossRefGoogle Scholar
13McLaughlin, D., Papanicolaou, G., Sulem, C. and Sulem, P.-L.. Focusing singularity of cubic Schrödinger equation. Phys. Rev. A 34 (1986), 1200–1210.CrossRefGoogle ScholarPubMed
14Merle, F.. Limit of the solution of the nonlinear Schrödinger equation at the blow-up time. J. Fund. Anal. 84 (1989), 201–214.CrossRefGoogle Scholar
15Merle, F.. Construction of solutions with exactly k blow-up points for the Schrodinger equation with critical nonlinearity. Comm. math. Phys. 129 (1990), 223–240.CrossRefGoogle Scholar
16Merle, F. and Tsutsumi, Y.. L2 concentratin of blow-up solutions for the nonlinear Schrodinger equation with the critical power nonlinearity. J. Differential Equations. 84 (1990), 205–214.CrossRefGoogle Scholar
17Ni, W.-M. and Serrin, J.. Nonexistence theroems for singular solutions of quasilinear partial differential equations. Comm. Pure Appl. Math. 39 (1986), 379–399.CrossRefGoogle Scholar
18Strauss, W. A.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
19Peletier, L. A. and Serrin, J.. Uniqueness of solutions of semilinear equations in ℝn. J. Differential Equations 61 (1986), 380–397.CrossRefGoogle Scholar
20Pan, X.-B. and Wang, X.-F.. Blow-up behavior of ground states of a semilinear elliptic equation in ℝn involving critical Sobolev exponent. J. Differential Equations (to appear).Google Scholar
21Tsutsumi, M.. Nonexistence and instability of solutions of nonlinear Schrödinger equations (unpublished).Google Scholar
22Tsutsumi, Y.. Rate of L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power. Nonlinear Anal. 15 (1990), 719–724.CrossRefGoogle Scholar
23Weinstein, M. I.. On the structure and formation of singularities in solutions to nonlinear dispersive evolution equation. Comm. Partial Differential Equations 11 (1986), 545–565.CrossRefGoogle Scholar