American Institute of Mathematical Sciences

Advanced
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.  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.  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.  doi: 10.5802/afst.600.  Google Scholar

[4]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Math., (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.   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.   Google Scholar

[7]

K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian,, Probablility and Mathematical Statistics, 20 (2000), 293.   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, (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).  doi: 10.1155/2013/240863.  Google Scholar

[10]

D. Bors, Stability of nonlinear Dirichlet BVPs governed by fractional Laplacian,, Real World Scientific, 2014 (2014).  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.  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.  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.  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.  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.  doi: 10.4064/bc66-0-8.  Google Scholar

[17]

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones,, Academic Press, (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.  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, (1980), 220.   Google Scholar

[20]

M. A. Krasnosielski, Topological Methods in the Theory of Nonlinear Integral Equations,, translated by A. H. Armstrong, (1964).   Google Scholar

[21]

T. Kulczycki, Gradient estimates of q-harmonic functions of fractional Schrödinger operator,, Potential Analysis, 39 (2013), 69.  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.   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.  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.  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.  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.  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.  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.   Google Scholar

[29]

R. Stańczy, Nonlocal elliptic equations,, Nonlinear Analysis, 47 (2001), 3579.  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.  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.  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.   Google Scholar

[33]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators,, Nonlinear Partial Differential Equations, 7 (2012), 271.   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.  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.  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.  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.  doi: 10.5802/afst.600.  Google Scholar

[4]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Math., (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.   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.   Google Scholar

[7]

K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian,, Probablility and Mathematical Statistics, 20 (2000), 293.   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, (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).  doi: 10.1155/2013/240863.  Google Scholar

[10]

D. Bors, Stability of nonlinear Dirichlet BVPs governed by fractional Laplacian,, Real World Scientific, 2014 (2014).  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.  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.  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.  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.  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.  doi: 10.4064/bc66-0-8.  Google Scholar

[17]

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones,, Academic Press, (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.  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, (1980), 220.   Google Scholar

[20]

M. A. Krasnosielski, Topological Methods in the Theory of Nonlinear Integral Equations,, translated by A. H. Armstrong, (1964).   Google Scholar

[21]

T. Kulczycki, Gradient estimates of q-harmonic functions of fractional Schrödinger operator,, Potential Analysis, 39 (2013), 69.  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.   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.  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.  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.  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.  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.  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.   Google Scholar

[29]

R. Stańczy, Nonlocal elliptic equations,, Nonlinear Analysis, 47 (2001), 3579.  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.  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.  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.   Google Scholar

[33]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators,, Nonlinear Partial Differential Equations, 7 (2012), 271.   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.  doi: 10.1007/BF00250743.  Google Scholar

[1]

Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[2]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[3]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[4]

Mei Yu, Xia Zhang, Binlin Zhang. Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3597-3612. doi: 10.3934/cpaa.2020157
[5]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[6]

Vincenzo Ambrosio. Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 781-815. doi: 10.3934/dcds.2020062
[7]

Leszek Gasiński, Nikolaos S. Papageorgiou. Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1421-1437. doi: 10.3934/cpaa.2009.8.1421
[8]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[9]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253
[10]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[11]

Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008
[12]

Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3183-3201. doi: 10.3934/dcds.2015.35.3183
[13]

Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099
[14]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[15]

Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026
[16]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[17]

Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253
[18]

De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431
[19]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
[20]

Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences