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$L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials
Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian
1. | Department of Mathematics, EPFL SB CAMA, Station 8 CH-1015 Lausanne, Switzerland |
2. | Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy |
We consider a class of parametric Schrödinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. Using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.
References:
[1] |
C. O. Alves,
Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206.
doi: 10.1016/S0362-546X(01)00887-2. |
[2] |
C. O. Alves and V. Ambrosio,
A multiplicity result for a nonlinear fractional Schrödinger equation in $\mathbb{R}^{N}$ without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 466 (2018), 498-522.
doi: 10.1016/j.jmaa.2018.06.005. |
[3] |
C. O. Alves and G. M. Figueiredo,
Existence and multiplicity of positive solutions to a p-Laplacian equation in $\mathbb{R}^{N}$, Differential Integral Equations, 19 (2006), 143-162.
|
[4] |
C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp.
doi: 10.1007/s00526-016-0983-x. |
[5] |
V. Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential, Electron. J. Diff. Equ., 2016 (2016), Paper No. 151, 12 pp. |
[6] |
V. Ambrosio,
Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl.(4), 196 (2017), 2043-2062.
doi: 10.1007/s10231-017-0652-5. |
[7] |
V. Ambrosio,
Concentration phenomena for critical fractional Schrödinger systems, Commun. Pure Appl. Anal., 17 (2018), 2085-2123.
doi: 10.3934/cpaa.2018099. |
[8] |
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$,
Rev. Mat. Iberoam., (in press), arXiv:1612.02388. |
[9] |
V. Ambrosio and G. M. Figueiredo,
Ground state solutions for a fractional Schrödinger equation with critical growth, Asymptot. Anal., 105 (2017), 159-191.
doi: 10.3233/ASY-171438. |
[10] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645.
|
[11] |
P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira,
Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Analysis, 164 (2017), 38-53.
doi: 10.1016/j.na.2017.08.005. |
[12] |
V. Benci and G. Cerami,
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.
doi: 10.1007/BF01234314. |
[13] |
L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations, 55 (2016), Art. 23, 32 pp.
doi: 10.1007/s00526-016-0958-y. |
[14] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[15] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[16] |
L. M. Del Pezzo and A. Quaas,
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations, 263 (2017), 765-778.
doi: 10.1016/j.jde.2017.02.051. |
[17] |
A. Di Castro, T. Kuusi and G. Palatucci,
Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.
doi: 10.1016/j.anihpc.2015.04.003. |
[18] |
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. |
[19] |
Y. Ding,
Variational Methods for Strongly Indefinite Problems, Interdisciplinary Mathematical Sciences, 7. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. ⅷ+168 pp.
doi: 10.1142/9789812709639. |
[20] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. ⅷ+152 pp.
doi: 10.1007/978-88-7642-601-8. |
[21] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[22] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[23] |
G. M. Figueiredo,
Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems with critical growth, Comm. Appl. Nonlinear Anal., 13 (2006), 79-99.
|
[24] |
G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $\mathbb{R}^{N}$,
NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 12, 22 pp.
doi: 10.1007/s00030-016-0355-4. |
[25] |
A. Fiscella and P. Pucci,
Kirchhoff-Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud., 17 (2017), 429-456.
doi: 10.1515/ans-2017-6021. |
[26] |
G. Franzina and G. Palatucci,
Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[27] |
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. |
[28] |
T. Isernia,
Positive solution for nonhomogeneous sublinear fractional equations in $\mathbb{R}^{N}$, Complex Var. Elliptic Equ., 63 (2018), 689-714.
doi: 10.1080/17476933.2017.1332052. |
[29] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[30] |
N. Laskin, Fractional Schrödinger equation,
Phys. Rev. E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[31] |
E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[32] |
J. Mawhin and M. Willem,
Critical Point Theory and Hamiltonian Systems, Springer-Verlag, 1989.
doi: 10.1007/978-1-4757-2061-7. |
[33] |
C. Mercuri and M. Willem,
A global compactness result for the p-Laplacian involving critical nonlinearities, Discrete Contin. Dyn. Syst., 28 (2010), 469-493.
doi: 10.3934/dcds.2010.28.469. |
[34] |
G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162, Cambridge, 2016.
doi: 10.1017/CBO9781316282397. |
[35] |
S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Brezis-Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations, 55 (2016), Art. 105, 25 pp.
doi: 10.1007/s00526-016-1035-2. |
[36] |
J. Moser,
A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.
doi: 10.1002/cpa.3160130308. |
[37] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[38] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[39] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$,
J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[40] |
X. Shang, J. Zhang and Y. Yang, On fractional Schrödinger equations with critical growth, J. Math. Phys., 54 (2013), 121502, 20 pp.
doi: 10.1063/1.4835355. |
[41] |
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, edited by D. Y. Gao and D. Montreanu, International Press, Boston, 2010,597–632. |
[42] |
J. Zhang,
Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.
doi: 10.1007/PL00001512. |
show all references
References:
[1] |
C. O. Alves,
Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206.
doi: 10.1016/S0362-546X(01)00887-2. |
[2] |
C. O. Alves and V. Ambrosio,
A multiplicity result for a nonlinear fractional Schrödinger equation in $\mathbb{R}^{N}$ without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 466 (2018), 498-522.
doi: 10.1016/j.jmaa.2018.06.005. |
[3] |
C. O. Alves and G. M. Figueiredo,
Existence and multiplicity of positive solutions to a p-Laplacian equation in $\mathbb{R}^{N}$, Differential Integral Equations, 19 (2006), 143-162.
|
[4] |
C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp.
doi: 10.1007/s00526-016-0983-x. |
[5] |
V. Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential, Electron. J. Diff. Equ., 2016 (2016), Paper No. 151, 12 pp. |
[6] |
V. Ambrosio,
Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl.(4), 196 (2017), 2043-2062.
doi: 10.1007/s10231-017-0652-5. |
[7] |
V. Ambrosio,
Concentration phenomena for critical fractional Schrödinger systems, Commun. Pure Appl. Anal., 17 (2018), 2085-2123.
doi: 10.3934/cpaa.2018099. |
[8] |
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$,
Rev. Mat. Iberoam., (in press), arXiv:1612.02388. |
[9] |
V. Ambrosio and G. M. Figueiredo,
Ground state solutions for a fractional Schrödinger equation with critical growth, Asymptot. Anal., 105 (2017), 159-191.
doi: 10.3233/ASY-171438. |
[10] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645.
|
[11] |
P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira,
Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Analysis, 164 (2017), 38-53.
doi: 10.1016/j.na.2017.08.005. |
[12] |
V. Benci and G. Cerami,
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.
doi: 10.1007/BF01234314. |
[13] |
L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations, 55 (2016), Art. 23, 32 pp.
doi: 10.1007/s00526-016-0958-y. |
[14] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[15] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[16] |
L. M. Del Pezzo and A. Quaas,
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations, 263 (2017), 765-778.
doi: 10.1016/j.jde.2017.02.051. |
[17] |
A. Di Castro, T. Kuusi and G. Palatucci,
Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.
doi: 10.1016/j.anihpc.2015.04.003. |
[18] |
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. |
[19] |
Y. Ding,
Variational Methods for Strongly Indefinite Problems, Interdisciplinary Mathematical Sciences, 7. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. ⅷ+168 pp.
doi: 10.1142/9789812709639. |
[20] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. ⅷ+152 pp.
doi: 10.1007/978-88-7642-601-8. |
[21] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[22] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[23] |
G. M. Figueiredo,
Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems with critical growth, Comm. Appl. Nonlinear Anal., 13 (2006), 79-99.
|
[24] |
G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $\mathbb{R}^{N}$,
NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 12, 22 pp.
doi: 10.1007/s00030-016-0355-4. |
[25] |
A. Fiscella and P. Pucci,
Kirchhoff-Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud., 17 (2017), 429-456.
doi: 10.1515/ans-2017-6021. |
[26] |
G. Franzina and G. Palatucci,
Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[27] |
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. |
[28] |
T. Isernia,
Positive solution for nonhomogeneous sublinear fractional equations in $\mathbb{R}^{N}$, Complex Var. Elliptic Equ., 63 (2018), 689-714.
doi: 10.1080/17476933.2017.1332052. |
[29] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[30] |
N. Laskin, Fractional Schrödinger equation,
Phys. Rev. E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[31] |
E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[32] |
J. Mawhin and M. Willem,
Critical Point Theory and Hamiltonian Systems, Springer-Verlag, 1989.
doi: 10.1007/978-1-4757-2061-7. |
[33] |
C. Mercuri and M. Willem,
A global compactness result for the p-Laplacian involving critical nonlinearities, Discrete Contin. Dyn. Syst., 28 (2010), 469-493.
doi: 10.3934/dcds.2010.28.469. |
[34] |
G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162, Cambridge, 2016.
doi: 10.1017/CBO9781316282397. |
[35] |
S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Brezis-Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations, 55 (2016), Art. 105, 25 pp.
doi: 10.1007/s00526-016-1035-2. |
[36] |
J. Moser,
A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.
doi: 10.1002/cpa.3160130308. |
[37] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[38] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[39] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$,
J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[40] |
X. Shang, J. Zhang and Y. Yang, On fractional Schrödinger equations with critical growth, J. Math. Phys., 54 (2013), 121502, 20 pp.
doi: 10.1063/1.4835355. |
[41] |
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, edited by D. Y. Gao and D. Montreanu, International Press, Boston, 2010,597–632. |
[42] |
J. Zhang,
Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.
doi: 10.1007/PL00001512. |
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