Open maps: Small and large holes with unusual properties
$L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials
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
Vincenzo Ambrosio 1, and Teresa Isernia 2,
Department of Mathematics, EPFL SB CAMA, Station 8 CH-1015 Lausanne, Switzerland
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.
Keywords: Fractional Schrödinger equation, fractional $p$-Laplacian operator, Nehari manifold, Ljusternik-Schnirelmann theory, critical growth.
Mathematics Subject Classification: Primary: 47G20, 35R11; Secondary: 35A15, 58E05.
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
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
V. Ambrosio, Concentration phenomena for critical fractional Schrödinger systems, Commun. Pure Appl. Anal., 17 (2018), 2085-2123. doi: 10.3934/cpaa.2018099. Google Scholar
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, Rev. Mat. Iberoam., (in press), arXiv:1612.02388. Google Scholar
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. Google Scholar
V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386. Google Scholar
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. Google Scholar
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. Google Scholar
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar
E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1. Google Scholar
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2061-7. Google Scholar
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. Google Scholar
G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162, Cambridge, 2016. doi: 10.1017/CBO9781316282397. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108
Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285
Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419
Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499
Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298
Hassan Emamirad, Arnaud Rougirel. Feynman path formula for the time fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020246
Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813
Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 519-538. doi: 10.3934/cpaa.2019026
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Vincenzo Ambrosio Teresa Isernia