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

Existence and multiplicity results for resonant fractional boundary value problems

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.

Citation: Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete and 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.

[2]

B. BarriosE. ColoradoR. 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.

[3]

T. BartschA. 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.

[4]

Z. BinlinG. 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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

E. Di NezzaG. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

A. IannizzottoS. LiuK. 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.

[22]

A. IannizzottoS. 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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

K. PereraM. 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.

[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.

[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.

[33]

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

[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.

[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.

[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.

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.

[2]

B. BarriosE. ColoradoR. 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.

[3]

T. BartschA. 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.

[4]

Z. BinlinG. 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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

E. Di NezzaG. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

A. IannizzottoS. LiuK. 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.

[22]

A. IannizzottoS. 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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

K. PereraM. 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.

[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.

[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.

[33]

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

[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.

[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.

[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.

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