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

Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian

Tadeusz Kulczycki 1, and  Robert Stańczy 2,

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.
Keywords: elliptic equation, fractional Laplacian, Superlinear BVPs, multiple solutions..
Mathematics Subject Classification: Primary: 35Q, 35J66, 35R1.
Citation: Tadeusz Kulczycki, Robert Stańczy. Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Cambridge Univ. Press, 1996.  Google Scholar

[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.  Google Scholar

[6]

K. Bogdan and T. Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domain, Studia Math., 133 (1999), 53-92.  Google Scholar

[7]

K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probablility and Mathematical Statistics, 20 (2000), 293-335.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, FL, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations, Colloq. Math., 92 (2002), 141-151. doi: 10.4064/cm92-1-12.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. Stańczy, Hammerstein equations with an integral over a non-compact domain, Annales Polonici Mathematici, 69 (1998), 49-60.  Google Scholar

[29]

R. Stańczy, Nonlocal elliptic equations, Nonlinear Analysis, 47 (2001), 3579-3584. doi: 10.1016/S0362-546X(01)00478-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.  Google Scholar

[33]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 271-298. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Cambridge Univ. Press, 1996.  Google Scholar

[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.  Google Scholar

[6]

K. Bogdan and T. Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domain, Studia Math., 133 (1999), 53-92.  Google Scholar

[7]

K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probablility and Mathematical Statistics, 20 (2000), 293-335.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, FL, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations, Colloq. Math., 92 (2002), 141-151. doi: 10.4064/cm92-1-12.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. Stańczy, Hammerstein equations with an integral over a non-compact domain, Annales Polonici Mathematici, 69 (1998), 49-60.  Google Scholar

[29]

R. Stańczy, Nonlocal elliptic equations, Nonlinear Analysis, 47 (2001), 3579-3584. doi: 10.1016/S0362-546X(01)00478-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.  Google Scholar

[33]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 271-298. Google Scholar

[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.  Google Scholar

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