November 2018, 38(11): 5835-5881. doi: 10.3934/dcds.2018254

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

Received  February 2018 Revised  July 2018 Published  August 2018

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.

Citation: Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254
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. BelchiorH. BuenoO. 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 CastroT. 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 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.

[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. FelmerA. 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. 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.

[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. BelchiorH. BuenoO. 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 CastroT. 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 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.

[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. FelmerA. 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. 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.

[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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