Research and Innovation Articles
Published 2019-02-19
Keywords
- Elliptic equations,
- Random potentials,
- Random nonlinear equations
How to Cite
Cioletti, L., Ferreira, L. C. F., & Furtado, M. (2019). An elliptic equation with random potential and supercritical nonlinearity. Revista Integración, Temas De matemáticas, 37(1), 1–16. https://doi.org/10.18273/revint.v37n1-2019001
Abstract
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.
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References
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[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), No. 4, 539–552.
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[2] Ambrosetti A., Malchiodi A. and Secchi S., “Multiplicity results for some nonlinear Schrödinger equations with potentials”, Arch. Ration. Mech. Anal. 159 (2001), No.3, 253–271.
[3] Bal G., Komorowski T. and Ryzhik L., “Asymptotic of the Solutions of the Random Schrödinger Equation”, Arch. Ration. Mech. Anal. 200 (2011), No.2, 613–664.
[4] Beck L. and Flandoli F., “A regularity theorem for quasilinear parabolic systems under random perturbations”, J. Evol. Equ. 13 (2013), No. 4, 829–874.
[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), 153–225.
[7] Dawson D.A. and Kouritzin M., “Invariance Principles for Parabolic Equations with Random Coefficients”, J. Funct. Anal. 149 (1997), No. 2, 377–414.
[8] Del Pino M. and Felmer P., “Local Mountain Pass for semilinear elliptic problems in unbounded domains”, Calc. Var. Partial Differential Equations 4 (1996), No. 2, 121–137.
[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 derivative terms”, Milan J. Math. 85 (2017), No. 2, 187–213.
[12] Ferreira L.C.F. and Mesquita C.A.A.S., “Existence and symmetries for elliptic equations with multipolar potentials and polyharmonic operators”, Indiana Univ. Math. J. 62 (2013), No. 6, 1955–1982.
[13] Ferreira L.C.F. and Montenegro M., “A Fourier approach for nonlinear equations with singular data”, Israel J. Math. 193 (2013), No. 1, 83–107.
[14] Ferreira L.C.F. and Montenegro M., “Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials”, J. Differential Equations 250 (2011), No.4, 2045–2063.
[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), No. 4, 539–552.
[16] Flandoli F., “Random perturbation of PDEs and fluid dynamic models”, in Saint Flour Summer School Lectures 2010, Lecture Notes in Math., Springer (2011), Heidelberg, 1–176.
[17] Hille E. and Phillips R.S., Functional Analysis and Semigroups, Amer. Math Soc. Colloquium Publ. 31, Amer. Math. Soc., Providence, Rhode, 1957.
[18] KirschW., “An invitation to random Schrödinger operators”, in Random Schrödinger operators, Vol. 25 of Panor. Synthèses, Soc. Math. France , Paris, (2008), 1-119.
[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.