- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 295480, 9 pages
Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains
1Mathematics Department, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Mathematics Department, College of Sciences and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia
Received 20 February 2013; Accepted 10 July 2013
Academic Editor: Daniel C. Biles
Copyright © 2013 Imed Bachar and Habib Mâagli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- N. S. Landkof, Foundations of Modern Potential Theory, Springer, New York, NY, USA, 1972.
- E. M. Stein, Singular integrals and differentiability properties of functions, vol. 30 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1970.
- D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2nd edition, 2009.
- J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1996.
- E. Valdinoci, “From the long jump random walk to the fractional Laplacian,” Boletín de la Sociedad Española de Matemática Aplicada, no. 49, pp. 33–44, 2009.
- L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, “Nonlocal models for nonlinear, dispersive waves,” Physica D, vol. 40, no. 3, pp. 360–392, 1989.
- K. Bogdan and T. Byczkowski, “Potential theory for the -stable Schrödinger operator on bounded Lipschitz domains,” Studia Mathematica, vol. 133, no. 1, pp. 53–92, 1999.
- K. Bogdan, “Representation of -harmonic functions in Lipschitz domains,” Hiroshima Mathematical Journal, vol. 29, no. 2, pp. 227–243, 1999.
- L. Caffarelli and L. Silvestre, “An extension problem related to the fractional Laplacian,” Communications in Partial Differential Equations, vol. 32, no. 7-9, pp. 1245–1260, 2007.
- A. Elgart and B. Schlein, “Mean field dynamics of boson stars,” Communications on Pure and Applied Mathematics, vol. 60, no. 4, pp. 500–545, 2007.
- J. Fröhlich and E. Lenzmann, “Blowup for nonlinear wave equations describing boson stars,” Communications on Pure and Applied Mathematics, vol. 60, no. 11, pp. 1691–1705, 2007.
- L. A. Caffarelli, “Further regularity for the Signorini problem,” Communications in Partial Differential Equations, vol. 4, no. 9, pp. 1067–1075, 1979.
- A. Signorini, “Questioni di elasticità non linearizzata e semilinearizzata,” vol. 18, pp. 95–139, 1959.
- L. A. Caffarelli and A. Vasseur, “Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,” Annals of Mathematics, vol. 171, no. 3, pp. 1903–1930, 2010.
- I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, “The structure of the free boundary for lower dimensional obstacle problems,” American Journal of Mathematics, vol. 130, no. 2, pp. 485–498, 2008.
- L. A. Caffarelli, S. Salsa, and L. Silvestre, “Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian,” Inventiones Mathematicae, vol. 171, no. 2, pp. 425–461, 2008.
- M. Kassmann, “A priori estimates for integro-differential operators with measurable kernels,” Calculus of Variations and Partial Differential Equations, vol. 34, no. 1, pp. 1–21, 2009.
- L. Silvestre, “Regularity of the obstacle problem for a fractional power of the Laplace operator,” Communications on Pure and Applied Mathematics, vol. 60, no. 1, pp. 67–112, 2007.
- S. Abe and S. Thurner, “Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion,” Physica A, vol. 356, no. 2-4, pp. 403–407, 2005.
- M. Jara, “Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,” Communications on Pure and Applied Mathematics, vol. 62, no. 2, pp. 198–214, 2009.
- A. Mellet, S. Mischler, and C. Mouhot, “Fractional diffusion limit for collisional kinetic equations,” Archive for Rational Mechanics and Analysis, vol. 199, no. 2, pp. 493–525, 2011.
- L. Vlahos, H. Isliker, Y. Kominis, and K. Hizonidis, “Normal and anomalous diffusion: a tutorial,” in Order and Chaos, T. Bountis, Ed., vol. 10, Patras University Press, Patras, Greece, 2008.
- H. Weitzner and G. M. Zaslavsky, “Some applications of fractional equations. Chaotic transport and complexity in classical and quantum dynamics,” Communications in Nonlinear Science and Numerical Simulation, vol. 8, no. 3-4, pp. 273–281, 2003.
- J. L. Vazquez, “Nonlinear diffusion with fractional Laplacian operators,” Non-Linear Partial Differential Equations, vol. 7, pp. 271–298, 2012.
- E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker's guide to the fractional Sobolev spaces,” Bulletin des Sciences Mathématiques, vol. 136, no. 5, pp. 521–573, 2012.
- R. Chemmam, H. Mâagli, and S. Masmoudi, “On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 5, pp. 1555–1576, 2011.
- H. Mâagli and M. Zribi, “On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains of ,” Positivity, vol. 9, no. 4, pp. 667–686, 2005.
- R. Chemmam, H. Mâagli, and S. Masmoudi, “Boundary behavior of positive solutions of a semilinear fractional Dirichlet problem,” Journal of Abstract Differential Equations and Applications, vol. 3, no. 2, pp. 75–90, 2012.
- F. C. Cîrstea and V. Rădulescu, “Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,” Asymptotic Analysis, vol. 46, no. 3-4, pp. 275–298, 2006.
- V. Marić, Regular Variation and Differential Equations, vol. 1726 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
- E. Seneta, Regularly Varying Functions, vol. 508, Springer, Berlin, Germany, 1976.
- S. Gontara, H. Mâagli, S. Masmoudi, and S. Turki, “Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 369, no. 2, pp. 719–729, 2010.
- H. Mâagli, “Asymptotic behavior of positive solutions of a semilinear Dirichlet problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 9, pp. 2941–2947, 2011.
- A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” Journal of Functional Analysis, vol. 122, no. 2, pp. 519–543, 1994.
- L. Boccardo, “A Dirichlet problem with singular and supercritical nonlinearities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 12, pp. 4436–4440, 2012.
- R. Chemmam, H. Mâagli, S. Masmoudi, and M. Zribi, “Combined effects in nonlinear singular elliptic problems in a bounded domain,” Advances in Nonlinear Analysis, vol. 1, no. 4, pp. 301–318, 2012.
- V. Rădulescu and D. Repovš, “Combined effects in nonlinear problems arising in the study of anisotropic continuous media,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 3, pp. 1524–1530, 2012.
- S. Yijing and L. Shujie, “Some remarks on a superlinear-singular problem: estimates of ,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2636–2650, 2008.
- Y. Sun, S. Wu, and Y. Long, “Combined effects of singular and superlinear nonlinearities in some singular boundary value problems,” Journal of Differential Equations, vol. 176, no. 2, pp. 511–531, 2001.
- Z.-Q. Chen and R. Song, “Estimates on Green functions and Poisson kernels for symmetric stable processes,” Mathematische Annalen, vol. 312, no. 3, pp. 465–501, 1998.
- T. Kulczycki, “Properties of Green function of symmetric stable processes,” Probability and Mathematical Statistics, vol. 17, pp. 339–364, 1997.