Existence and multiplicity results for resonant fractional boundary value problems

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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 and Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028

References:

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[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.
[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.
[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.
[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 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.
[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. 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.
[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.
[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.
[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. 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.
[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. 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.
[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.
[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.
[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 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.
[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. 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.
[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.
[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.
[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. 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.
[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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