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Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity
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Some remarks on boundary operators of Bessel extensions
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 |
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.
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Elliptic problems involving the fractional Laplacian in ${\mathbb R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
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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.
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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. |
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Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.
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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. |
[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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