American Institute of Mathematical Sciences

December  2015, 35(12): 5555-5607. doi: 10.3934/dcds.2015.35.5555

Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian

 1 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, UFR des Sciences, 33, rue Saint-Leu, 80039, Amiens Cedex 1, France

Received  November 2013 Revised  March 2014 Published  May 2015

We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems of the form $$\left\lbrace\begin{array}{ll} (-\triangle)^s u = \pm\,f(x,u) & \hbox{ in }\Omega \\ u=g & \hbox{ in }\mathbb{R}^n\setminus\overline{\Omega}\\ Eu=h & \hbox{ on }\partial\Omega \end{array}\right.$$ when the nonlinearity $f$ and the boundary data $g,h$ are positive, but allowing the right-hand side to be both positive or negative and looking for solutions that blow up at the boundary. The operator $E$ is a weighted limit to the boundary: for example, if $\Omega$ is the ball $B$, there exists a constant $C(n,s)>0$ such that $$Eu(\theta) = C(n,s) \lim_{x \to \theta}_{x\in B} u(x) {dist(x,\partial B)}^{1-s}, \hbox{ for all } \theta \in \partial B.$$ Our starting observation is the existence of $s$-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative right-hand side and as subsolutions in the positive case.
Citation: Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555
References:
 [1] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-8137-3. [2] C. Bandle, Asymptotic behavior of large solutions of elliptic equations, Analele Universităţii din Craiova. Seria Matematică-Informatică, 32 (2005), 1-8. [3] K. Bogdan, Representation of $\alpha$-harmonic functions in Lipschitz domains, Hiroshima Mathematical Journal, 29 (1999), 227-243. [4] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Mathematica, 123 (1997), 43-80. [5] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondraček, Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02141-1. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [7] H. Chen, P. Felmer and A. Quaas, Large solutions to elliptic equations involving the fractional Laplacian, Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, in press, (2014). doi: 10.1016/j.anihpc.2014.08.001. [8] H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, Journal of Differential Equations, 257 (2014), 1457-1486. doi: 10.1016/j.jde.2014.05.012. [9] Z.-Q. Chen, Multidimensional symmetric stable processes, The Korean Journal of Computational & Applied Mathematics, 6 (1999), 227-266. [10] P. Clément and G. Sweers, Getting a solution between sub- and supersolutions without monotone iteration, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 19 (1987), 189-194. [11] O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry, Journal of Differential Equations, 249 (2010), 931-964. doi: 10.1016/j.jde.2010.02.023. [12] O. Costin, L. Dupaigne and O. Goubet, Uniqueness of large solutions, Journal of Mathematical Analysis and Applications, 395 (2012), 806-812. doi: 10.1016/j.jmaa.2012.05.085. [13] J.-S. Dhersin and J.-F. Le Gall, Wiener's test for super-Brownian motion and the Brownian snake, Probability Theory and Related Fields, 108 (1997), 103-129. doi: 10.1007/s004400050103. [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [15] S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Advanced Nonlinear Studies, 7 (2007), 271-298. [16] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802. [17] P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptotic Analysis, 78 (2012), 123-144. [18] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. [19] K. H. Karlsen, F. Petitta and S. Ulusoy, A duality approach to the fractional Laplacian with measure data, Publicacions Matemàtiques, 55 (2011), 151-161. doi: 10.5565/PUBLMAT_55111_07. [20] J. B. Keller, On solutions of $\Delta u=f(u)$, Communications on Pure and Applied Mathematics, 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. [21] T. Klimsiak and A. Rozkosz, Dirichlet forms and semilinear elliptic equations with measure data, Journal of Functional Analysis, 265 (2013), 890-925. doi: 10.1016/j.jfa.2013.05.028. [22] N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York, 1972. [23] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, Journal of Evolution Equations, 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y. [24] M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter, Berlin/Boston, 2014. [25] M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proceedings of the American Mathematical Society, 136 (2008), 2429-2438. doi: 10.1090/S0002-9939-08-09231-9. [26] B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Memoirs of the American Mathematical Society, 168 (2004), xvi+121 pp. doi: 10.1090/memo/0798. [27] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific Journal of Mathematics, 7 (1957), 1641-1647. [28] M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. (Szeged), 9 (1938), 1-42. [29] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathématiques Pures et Appliquées (9), 101 (2014), 275-302. [30] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153. [31] G. Stampacchia, Équations Elliptiques du Second Ordre à Coefficients Discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965), Les Presses de l'Université de Montréal, Montreal, Québec, 1966.

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References:
 [1] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-8137-3. [2] C. Bandle, Asymptotic behavior of large solutions of elliptic equations, Analele Universităţii din Craiova. Seria Matematică-Informatică, 32 (2005), 1-8. [3] K. Bogdan, Representation of $\alpha$-harmonic functions in Lipschitz domains, Hiroshima Mathematical Journal, 29 (1999), 227-243. [4] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Mathematica, 123 (1997), 43-80. [5] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondraček, Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02141-1. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [7] H. Chen, P. Felmer and A. Quaas, Large solutions to elliptic equations involving the fractional Laplacian, Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, in press, (2014). doi: 10.1016/j.anihpc.2014.08.001. [8] H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, Journal of Differential Equations, 257 (2014), 1457-1486. doi: 10.1016/j.jde.2014.05.012. [9] Z.-Q. Chen, Multidimensional symmetric stable processes, The Korean Journal of Computational & Applied Mathematics, 6 (1999), 227-266. [10] P. Clément and G. Sweers, Getting a solution between sub- and supersolutions without monotone iteration, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 19 (1987), 189-194. [11] O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry, Journal of Differential Equations, 249 (2010), 931-964. doi: 10.1016/j.jde.2010.02.023. [12] O. Costin, L. Dupaigne and O. Goubet, Uniqueness of large solutions, Journal of Mathematical Analysis and Applications, 395 (2012), 806-812. doi: 10.1016/j.jmaa.2012.05.085. [13] J.-S. Dhersin and J.-F. Le Gall, Wiener's test for super-Brownian motion and the Brownian snake, Probability Theory and Related Fields, 108 (1997), 103-129. doi: 10.1007/s004400050103. [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [15] S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Advanced Nonlinear Studies, 7 (2007), 271-298. [16] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802. [17] P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptotic Analysis, 78 (2012), 123-144. [18] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. [19] K. H. Karlsen, F. Petitta and S. Ulusoy, A duality approach to the fractional Laplacian with measure data, Publicacions Matemàtiques, 55 (2011), 151-161. doi: 10.5565/PUBLMAT_55111_07. [20] J. B. Keller, On solutions of $\Delta u=f(u)$, Communications on Pure and Applied Mathematics, 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. [21] T. Klimsiak and A. Rozkosz, Dirichlet forms and semilinear elliptic equations with measure data, Journal of Functional Analysis, 265 (2013), 890-925. doi: 10.1016/j.jfa.2013.05.028. [22] N. S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York, 1972. [23] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, Journal of Evolution Equations, 3 (2003), 637-652. doi: 10.1007/s00028-003-0122-y. [24] M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter, Berlin/Boston, 2014. [25] M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proceedings of the American Mathematical Society, 136 (2008), 2429-2438. doi: 10.1090/S0002-9939-08-09231-9. [26] B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Memoirs of the American Mathematical Society, 168 (2004), xvi+121 pp. doi: 10.1090/memo/0798. [27] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific Journal of Mathematics, 7 (1957), 1641-1647. [28] M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. (Szeged), 9 (1938), 1-42. [29] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathématiques Pures et Appliquées (9), 101 (2014), 275-302. [30] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153. [31] G. Stampacchia, Équations Elliptiques du Second Ordre à Coefficients Discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965), Les Presses de l'Université de Montréal, Montreal, Québec, 1966.
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