American Institute of Mathematical Sciences

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June  2018, 11(3): 511-532. doi: 10.3934/dcdss.2018028

Existence and multiplicity results for resonant fractional boundary value problems

Antonio Iannizzotto 1,, and  Nikolaos S. Papageorgiou 2,

1. 

Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy
2. 

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

* Corresponding author: A. Iannizzotto.

Received  May 2017 Revised  August 2017 Published  October 2017

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.

Keywords: Fractional Laplacian, eigenvalue problems, Morse theory.
Mathematics Subject Classification: Primary: 35R11, 35P05; Secondary: 49F15.
Citation: Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028
References:
[1]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${\mathbb R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar

[2]

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

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Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.  doi: 10.1088/0951-7715/28/7/2247.  Google Scholar

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C. Bucur and E. Valdinoci, Non-local Diffusion and Applications Springer, New York, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

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X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincaré (C) Nonlinear Analysis, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[7]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

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L. Caffarelli, Nonlocal diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[9]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[10]

X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992.  doi: 10.1016/j.jde.2014.01.027.  Google Scholar

[11]

W. Cheng and S. Deng, The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities, Z.A.M.P., 66 (2015), 1387-1400.  doi: 10.1007/s00033-014-0486-6.  Google Scholar

[12]

D. G. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.  doi: 10.1080/03605309208820844.  Google Scholar

[13]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[14]

F. G. Düzgün and A. Iannizzotto, Three nontrivial solutions for nonlinear fractional Laplacian equations, Adv. Nonlinear Anal. DOI: 10.1515/anona-2016-0090. doi: 10.1515/anona-2016-009.  Google Scholar

[15]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.  Google Scholar

[16]

R. Fei, J. Zhang and C. Ma, Multiple solutions to fractional equations without the Ambrosetti-Rabinowitz condition, Electr. J. Diff. Equations 2017 (2017), 11 p.  Google Scholar

[17]

A. Fiscella, Saddle point solutions for non-local elliptic operators, Topol. Methods Nonlinear Anal., 44 (2014), 527-538.  doi: 10.12775/TMNA.2014.059.  Google Scholar

[18]

S. Goyal and K. Sreenadh, On the Fučík spectrum of non-local elliptic operators, Nonlinear Differ. Equ. Appl., 21 (2014), 567-588.  doi: 10.1007/s00030-013-0258-6.  Google Scholar

[19]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[20]

H. Hofer, A geometric description of of the neighborhood of a critical point given by the mountain-pass theorem, J. London Math. Soc., 31 (1985), 566-570.  doi: 10.1112/jlms/s2-31.3.566.  Google Scholar

[21]

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.  Google Scholar

[22]

A. Iannizzotto, S. Mosconi and M. Squassina, $H^s$ versus $C^0$-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.  doi: 10.1007/s00030-014-0292-z.  Google Scholar

[23]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.  Google Scholar

[24]

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059.  Google Scholar

[25]

Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.  doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar

[26]

G. Molica Bisci and V. D. Rădulescu, Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl., 22 (2015), 721-739.  doi: 10.1007/s00030-014-0302-1.  Google Scholar

[27]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar

[28]

D. Mugnai and D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111-124.  doi: 10.1515/acv-2015-0032.  Google Scholar

[29]

N. S. Papageorgiou and V. D. Rădulescu, Semilinear Robin problems resonant at both zero and infinity, Forum Math., 69 (2017), 261-286.  doi: 10.2748/tmj/1498269626.  Google Scholar

[30]

K. Perera, M. Squassina and Y. Yang, A note on the Dancer-Fučík spectra of the fractional $p$-Laplacian and Laplacian operators, Adv. Nonlinear Anal., 4 (2015), 13-23.  doi: 10.1515/anona-2014-0038.  Google Scholar

[31]

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

[32]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[33]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[34]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.  Google Scholar

[35]

K. Teng, Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators, Annali Mat. Pura Appl., 194 (2015), 1455-1468.  doi: 10.1007/s10231-014-0428-0.  Google Scholar

[36]

Y. Wei and X. Su, Multiplicity of solutions for nonlocal elliptic equations driven by the fractional Laplacian, Calc. Var., 52 (2015), 95-124.  doi: 10.1007/s00526-013-0706-5.  Google Scholar

show all references

References:
[1]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${\mathbb R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar

[2]

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

[3]

T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations, Handbook of Global Analysis, Elsevier, Amsterdam, 1211 (2008), 41-73.  doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar

[4]

Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.  doi: 10.1088/0951-7715/28/7/2247.  Google Scholar

[5]

C. Bucur and E. Valdinoci, Non-local Diffusion and Applications Springer, New York, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[6]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincaré (C) Nonlinear Analysis, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[7]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

[8]

L. Caffarelli, Nonlocal diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[9]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[10]

X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992.  doi: 10.1016/j.jde.2014.01.027.  Google Scholar

[11]

W. Cheng and S. Deng, The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities, Z.A.M.P., 66 (2015), 1387-1400.  doi: 10.1007/s00033-014-0486-6.  Google Scholar

[12]

D. G. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.  doi: 10.1080/03605309208820844.  Google Scholar

[13]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[14]

F. G. Düzgün and A. Iannizzotto, Three nontrivial solutions for nonlinear fractional Laplacian equations, Adv. Nonlinear Anal. DOI: 10.1515/anona-2016-0090. doi: 10.1515/anona-2016-009.  Google Scholar

[15]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.  Google Scholar

[16]

R. Fei, J. Zhang and C. Ma, Multiple solutions to fractional equations without the Ambrosetti-Rabinowitz condition, Electr. J. Diff. Equations 2017 (2017), 11 p.  Google Scholar

[17]

A. Fiscella, Saddle point solutions for non-local elliptic operators, Topol. Methods Nonlinear Anal., 44 (2014), 527-538.  doi: 10.12775/TMNA.2014.059.  Google Scholar

[18]

S. Goyal and K. Sreenadh, On the Fučík spectrum of non-local elliptic operators, Nonlinear Differ. Equ. Appl., 21 (2014), 567-588.  doi: 10.1007/s00030-013-0258-6.  Google Scholar

[19]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[20]

H. Hofer, A geometric description of of the neighborhood of a critical point given by the mountain-pass theorem, J. London Math. Soc., 31 (1985), 566-570.  doi: 10.1112/jlms/s2-31.3.566.  Google Scholar

[21]

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.  Google Scholar

[22]

A. Iannizzotto, S. Mosconi and M. Squassina, $H^s$ versus $C^0$-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.  doi: 10.1007/s00030-014-0292-z.  Google Scholar

[23]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.  Google Scholar

[24]

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059.  Google Scholar

[25]

Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.  doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar

[26]

G. Molica Bisci and V. D. Rădulescu, Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl., 22 (2015), 721-739.  doi: 10.1007/s00030-014-0302-1.  Google Scholar

[27]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar

[28]

D. Mugnai and D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111-124.  doi: 10.1515/acv-2015-0032.  Google Scholar

[29]

N. S. Papageorgiou and V. D. Rădulescu, Semilinear Robin problems resonant at both zero and infinity, Forum Math., 69 (2017), 261-286.  doi: 10.2748/tmj/1498269626.  Google Scholar

[30]

K. Perera, M. Squassina and Y. Yang, A note on the Dancer-Fučík spectra of the fractional $p$-Laplacian and Laplacian operators, Adv. Nonlinear Anal., 4 (2015), 13-23.  doi: 10.1515/anona-2014-0038.  Google Scholar

[31]

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

[32]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[33]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[34]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.  Google Scholar

[35]

K. Teng, Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators, Annali Mat. Pura Appl., 194 (2015), 1455-1468.  doi: 10.1007/s10231-014-0428-0.  Google Scholar

[36]

Y. Wei and X. Su, Multiplicity of solutions for nonlocal elliptic equations driven by the fractional Laplacian, Calc. Var., 52 (2015), 95-124.  doi: 10.1007/s00526-013-0706-5.  Google Scholar

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