An elliptic equation with random potential and supercritical nonlinearity

• Cioletti, L. ^[1] ; Ferreira, L. C. F. ^[2] ; Furtado, M. ^[1]
  1. [1] Universidade de Brasília

     Universidade de Brasília

     Brasil

     
  2. [2] Universidade Estadual de Campinas

     Universidade Estadual de Campinas

     Brasil

     
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 37, Nº. 1, 2019 (Ejemplar dedicado a: Revista Integración, temas de matemáticas), págs. 1-16
• Idioma: inglés
• Títulos paralelos:
  • Una ecuación elíptica con potencial aleatorio y no linealidad supercrítica
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Estamos interesados en una ecuación elíptica no homogénea con potencial aleatorio y no linealidad supercrítica. Obtenemos la existencia de solución casi seguramente para una clase de potenciales que incluye continuos y discretos. Además, proporcionamos una ley de grandes números para las soluciones obtenidas por conjuntos independientes y estimamos el valor esperado para sus normas L∞.

  • English

    We are concerned with a nonhomogeneous elliptic equation with random potential and supercritical nonlinearity. Existence of solution is obtained almost surely for a class of potentials that includes continuum and discrete ones. Also, we provide a law of larger numbers for the obtained solutions by independent ensembles and estimate the expected value for their L∞-norms.

• Referencias bibliográficas
  • Citas [1] Ambrosetti A., Badiale M. and Cingolani S., “Semiclassical states of nonlinear Schrödinger equations”, Arch. Ration. Mech. Anal....
  • [2] Ambrosetti A., Malchiodi A. and Secchi S., “Multiplicity results for some nonlinear Schrödinger equations with potentials”, Arch. Ration....
  • [3] Bal G., Komorowski T. and Ryzhik L., “Asymptotic of the Solutions of the Random Schrödinger Equation”, Arch. Ration. Mech. Anal. 200 (2011),...
  • [4] Beck L. and Flandoli F., “A regularity theorem for quasilinear parabolic systems under random perturbations”, J. Evol. Equ. 13 (2013),...
  • [5] Bourgain J., “Nonlinear Schrödinger Equation with a Random Potential”, Illinois J. Math. 50 (2006), No. 1-4, 183–188.
  • [6] Conlon J.G. and Naddaf A., “Green’s Functions for Elliptic and Parabolic Equations with Random Coefficients”, New York J. Math. 6 (2000),...
  • [7] Dawson D.A. and Kouritzin M., “Invariance Principles for Parabolic Equations with Random Coefficients”, J. Funct. Anal. 149 (1997), No....
  • [8] Del Pino M. and Felmer P., “Local Mountain Pass for semilinear elliptic problems in unbounded domains”, Calc. Var. Partial Differential...
  • [9] Evans L.C., Partial differential equations, Graduate Studies in Mathematics 19, Amer. Math. Soc., Providence, RI, 1998.
  • [10] Fannjiang A., “Self-Averaging Scaling Limits for Random Parabolic Waves”, Arch. Ratio. Mech. Anal. 175 (2008), No. 3, 343–387.
  • [11] Ferreira L.C.F. and Castañeda-Centurión N.F., “A Fourier analysis approach to elliptic equations with critical potentials and nonlinear...
  • [12] Ferreira L.C.F. and Mesquita C.A.A.S., “Existence and symmetries for elliptic equations with multipolar potentials and polyharmonic operators”,...
  • [13] Ferreira L.C.F. and Montenegro M., “A Fourier approach for nonlinear equations with singular data”, Israel J. Math. 193 (2013), No. 1,...
  • [14] Ferreira L.C.F. and Montenegro M., “Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials”, J....
  • [15] Ferreira L.C.F., Medeiros E.S. and Montenegro M., “A class of elliptic equations in anisotropic spaces”, Ann. Mat. Pura Appl. 192 (2013),...
  • [16] Flandoli F., “Random perturbation of PDEs and fluid dynamic models”, in Saint Flour Summer School Lectures 2010, Lecture Notes in Math.,...
  • [17] Hille E. and Phillips R.S., Functional Analysis and Semigroups, Amer. Math Soc. Colloquium Publ. 31, Amer. Math. Soc., Providence, Rhode,...
  • [18] KirschW., “An invitation to random Schrödinger operators”, in Random Schrödinger operators, Vol. 25 of Panor. Synthèses, Soc. Math. France...
  • [19] Parthasarathy K.R., Probability Measures on Metric Spaces, Academic Press, Providence, 1967.
  • [20] Rabinowitz P.H., “On a class of nonlinear Schrödinger equations”, Z. Angew Math. Phys. 43 (1992), No. 2, 270–291.
  • [21] Safronov O., “Absolutely continuous spectrum of one random elliptic operator”, J. Funct. Anal. 255 (2008), No. 3, 755–767.