    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 & Continuous Dynamical Systems, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555
##### References:
  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.  Google Scholar  C. Bandle, Asymptotic behavior of large solutions of elliptic equations, Analele Universităţii din Craiova. Seria Matematică-Informatică, 32 (2005), 1-8. Google Scholar  K. Bogdan, Representation of $\alpha$-harmonic functions in Lipschitz domains, Hiroshima Mathematical Journal, 29 (1999), 227-243. Google Scholar  K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Mathematica, 123 (1997), 43-80. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  Z.-Q. Chen, Multidimensional symmetric stable processes, The Korean Journal of Computational & Applied Mathematics, 6 (1999), 227-266. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.  Google Scholar  P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptotic Analysis, 78 (2012), 123-144. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter, Berlin/Boston, 2014. Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific Journal of Mathematics, 7 (1957), 1641-1647. Google Scholar  M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. (Szeged), 9 (1938), 1-42. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar

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##### References:
  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.  Google Scholar  C. Bandle, Asymptotic behavior of large solutions of elliptic equations, Analele Universităţii din Craiova. Seria Matematică-Informatică, 32 (2005), 1-8. Google Scholar  K. Bogdan, Representation of $\alpha$-harmonic functions in Lipschitz domains, Hiroshima Mathematical Journal, 29 (1999), 227-243. Google Scholar  K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Mathematica, 123 (1997), 43-80. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  Z.-Q. Chen, Multidimensional symmetric stable processes, The Korean Journal of Computational & Applied Mathematics, 6 (1999), 227-266. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.  Google Scholar  P. Felmer and A. Quaas, Boundary blow up solutions for fractional elliptic equations, Asymptotic Analysis, 78 (2012), 123-144. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter, Berlin/Boston, 2014. Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific Journal of Mathematics, 7 (1957), 1641-1647. Google Scholar  M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. (Szeged), 9 (1938), 1-42. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar
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