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The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion
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 |
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. |
show all references
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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