# American Institute of Mathematical Sciences

October  2014, 19(8): 2581-2591. doi: 10.3934/dcdsb.2014.19.2581

## Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian

 1 Instytut Matematyki i Informatyki, Politechnika Wroc lawska, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland 2 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław

Received  November 2013 Revised  January 2014 Published  August 2014

The existence of at least two solutions to superlinear integral equation in cone is proved using the Krasnosielskii Fixed Point Theorem. The result is applied to the Dirichlet BVPs with the fractional Laplacian.
Citation: Tadeusz Kulczycki, Robert Stańczy. Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2581-2591. doi: 10.3934/dcdsb.2014.19.2581
##### References:
 [1] R. P. Agarwal and D. O'Regan, Existence theorem for single and multiple solutions to singular positone boundary value problems, J. Differential Equations, 175 (2001), 393-414. doi: 10.1006/jdeq.2001.3975. [2] H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11 (1972), 346-384. doi: 10.1016/0022-1236(72)90074-2. [3] P. Baras, Non-unicité des solutions d'une equation d'évolution non-linéaire, Annales Faculté des Sciences Toulouse, 5 (1983), 287-302. doi: 10.5802/afst.600. [4] J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Cambridge Univ. Press, 1996. [5] R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc., 99 (1961), 540-554. [6] K. Bogdan and T. Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domain, Studia Math., 133 (1999), 53-92. [7] K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probablility and Mathematical Statistics, 20 (2000), 293-335. [8] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondracek, Potential Analysis of Stable Processes and its Extensions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02141-1. [9] D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstract and Applied Analysis, 2013 (2013), ID 240863, 10 pp. doi: 10.1155/2013/240863. [10] D. Bors, Stability of nonlinear Dirichlet BVPs governed by fractional Laplacian, Real World Scientific, 2014 (2014), ID 920537, 10 pp. doi: 10.1155/2014/920537. [11] C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. [12] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [13] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, to appear in Trans. Amer. Math. Soc. [14] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [15] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [16] P. Fijałkowski, B. Przeradzki and R. Stańczy, A nonlocal elliptic equation in a bounded domain, Banach Center Publications, 66 (2004), 127-133. doi: 10.4064/bc66-0-8. [17] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, FL, 1988. [18] K. S. Ha and Y. H. Lee, Existence of multiple positive solutions of singular boundary value problems, Nonlinear Analysis Theory Methods and Appl., 28 (1997), 1429-1438. doi: 10.1016/0362-546X(95)00231-J. [19] A. Haraux and F. B. Weissler, Non-unicité pour un probléme de Cauchy semi-linéaire, in Nonlinear Partial Differential Equations and their Applications, Collége de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math., {70}, Pitman, Boston, Massachussets, London, 1982, 220-233. [20] M. A. Krasnosielski, Topological Methods in the Theory of Nonlinear Integral Equations, translated by A. H. Armstrong, translation edited by J. Burlak, A Pergamon Press Book The Macmillan Co., New York, 1964. [21] T. Kulczycki, Gradient estimates of q-harmonic functions of fractional Schrödinger operator, Potential Analysis, 39 (2013), 69-98. doi: 10.1007/s11118-012-9322-9. [22] Y. H. Lee, An existence result of positive solutions for singular superlinear boundary value problems and its applications, J. Korean Math. Soc., 34 (1997), 247-255. [23] E. di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathömatiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [24] B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations, Colloq. Math., 92 (2002), 141-151. doi: 10.4064/cm92-1-12. [25] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [26] X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozhaev identity and nonexistence results, C. R. Math. Acad. Sci., 350 (2012), 505-508. doi: 10.1016/j.crma.2012.05.011. [27] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, Journal of Functional Analysis, 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. [28] R. Stańczy, Hammerstein equations with an integral over a non-compact domain, Annales Polonici Mathematici, 69 (1998), 49-60. [29] R. Stańczy, Nonlocal elliptic equations, Nonlinear Analysis, 47 (2001), 3579-3584. doi: 10.1016/S0362-546X(01)00478-3. [30] R. Stańczy, Positive solutions for superlinear elliptic equations, Journal of Mathematical Analysis and Applications, 283 (2003), 159-166. doi: 10.1016/S0022-247X(03)00265-8. [31] R. Stańczy, Multiple solutions for equations involving bilinear, coercive and compact forms with applications to differential equations, Journal of Mathematical Analysis and Applications, 405 (2013), 416-421. doi: 10.1016/j.jmaa.2013.04.021. [32] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. [33] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 271-298. [34] F. B. Weissler, Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation, Arch. Rational Mech. Anal., 91 (1985), 231-245. doi: 10.1007/BF00250743.

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##### References:
 [1] R. P. Agarwal and D. O'Regan, Existence theorem for single and multiple solutions to singular positone boundary value problems, J. Differential Equations, 175 (2001), 393-414. doi: 10.1006/jdeq.2001.3975. [2] H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11 (1972), 346-384. doi: 10.1016/0022-1236(72)90074-2. [3] P. Baras, Non-unicité des solutions d'une equation d'évolution non-linéaire, Annales Faculté des Sciences Toulouse, 5 (1983), 287-302. doi: 10.5802/afst.600. [4] J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Cambridge Univ. Press, 1996. [5] R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc., 99 (1961), 540-554. [6] K. Bogdan and T. Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domain, Studia Math., 133 (1999), 53-92. [7] K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probablility and Mathematical Statistics, 20 (2000), 293-335. [8] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondracek, Potential Analysis of Stable Processes and its Extensions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02141-1. [9] D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstract and Applied Analysis, 2013 (2013), ID 240863, 10 pp. doi: 10.1155/2013/240863. [10] D. Bors, Stability of nonlinear Dirichlet BVPs governed by fractional Laplacian, Real World Scientific, 2014 (2014), ID 920537, 10 pp. doi: 10.1155/2014/920537. [11] C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. [12] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [13] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, to appear in Trans. Amer. Math. Soc. [14] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [15] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [16] P. Fijałkowski, B. Przeradzki and R. Stańczy, A nonlocal elliptic equation in a bounded domain, Banach Center Publications, 66 (2004), 127-133. doi: 10.4064/bc66-0-8. [17] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, FL, 1988. [18] K. S. Ha and Y. H. Lee, Existence of multiple positive solutions of singular boundary value problems, Nonlinear Analysis Theory Methods and Appl., 28 (1997), 1429-1438. doi: 10.1016/0362-546X(95)00231-J. [19] A. Haraux and F. B. Weissler, Non-unicité pour un probléme de Cauchy semi-linéaire, in Nonlinear Partial Differential Equations and their Applications, Collége de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math., {70}, Pitman, Boston, Massachussets, London, 1982, 220-233. [20] M. A. Krasnosielski, Topological Methods in the Theory of Nonlinear Integral Equations, translated by A. H. Armstrong, translation edited by J. Burlak, A Pergamon Press Book The Macmillan Co., New York, 1964. [21] T. Kulczycki, Gradient estimates of q-harmonic functions of fractional Schrödinger operator, Potential Analysis, 39 (2013), 69-98. doi: 10.1007/s11118-012-9322-9. [22] Y. H. Lee, An existence result of positive solutions for singular superlinear boundary value problems and its applications, J. Korean Math. Soc., 34 (1997), 247-255. [23] E. di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathömatiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [24] B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations, Colloq. Math., 92 (2002), 141-151. doi: 10.4064/cm92-1-12. [25] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [26] X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozhaev identity and nonexistence results, C. R. Math. Acad. Sci., 350 (2012), 505-508. doi: 10.1016/j.crma.2012.05.011. [27] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, Journal of Functional Analysis, 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. [28] R. Stańczy, Hammerstein equations with an integral over a non-compact domain, Annales Polonici Mathematici, 69 (1998), 49-60. [29] R. Stańczy, Nonlocal elliptic equations, Nonlinear Analysis, 47 (2001), 3579-3584. doi: 10.1016/S0362-546X(01)00478-3. [30] R. Stańczy, Positive solutions for superlinear elliptic equations, Journal of Mathematical Analysis and Applications, 283 (2003), 159-166. doi: 10.1016/S0022-247X(03)00265-8. [31] R. Stańczy, Multiple solutions for equations involving bilinear, coercive and compact forms with applications to differential equations, Journal of Mathematical Analysis and Applications, 405 (2013), 416-421. doi: 10.1016/j.jmaa.2013.04.021. [32] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. [33] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 271-298. [34] F. B. Weissler, Asymptotic analysis of an ordinary differential equation and nonuniqueness for a semilinear partial differential equation, Arch. Rational Mech. Anal., 91 (1985), 231-245. doi: 10.1007/BF00250743.
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