June  2022, 21(6): 1953-1968. doi: 10.3934/cpaa.2021111

Ground state solutions for the fractional problems with dipole-type potential and critical exponent

1. 

School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui 232001, China

2. 

School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

* Corresponding author

Dedicated to Professor Goong Chen on the occasion of his seventieth birthday

Received  December 2020 Revised  May 2021 Published  June 2022 Early access  June 2021

Fund Project: This work is supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0292)

We are concerned with ground state solutions of the fractional problems with dipole-type potential and critical exponent. Under certain conditions on the dipole-type potential and the parameter, we show that the structure of the Palais-Smale sequence goes to zero weakly, and establish the existence of ground state solution to the above problems by using a new analytical method not involving the concentration-compactness principle.

Citation: Yu Su, Zhaosheng Feng. Ground state solutions for the fractional problems with dipole-type potential and critical exponent. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1953-1968. doi: 10.3934/cpaa.2021111
References:
[1]

O. BourgetM. Courdurier and C. Fernandez, Construction of solutions for some localized nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 39 (2019), 841-862.  doi: 10.3934/dcds.2019035.

[2]

L. Caffarelli, Non-local diffusions, drifts and games, pp. 37-52 in "Nonlinear Partial Differential Equations" edt by H. Holden and K. Karlsen, Abel Symp., vol. 7, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25361-4.

[3]

A. Cotsiolis and N. K. Travoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[4]

P. d'AveniaG. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[5]

S. Dipierro, L. Montoro, I. Peral, B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations, 55 (2016), Art. 99. doi: 10.1007/s00526-016-1032-5.

[6]

M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.

[7]

T. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal., 268 (2015), 3278-3289.  doi: 10.1016/j.jfa.2015.03.016.

[8]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176.  doi: 10.1016/j.jfa.2013.08.027.

[9]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[10]

E. H. Lieb and H. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174. 

[11]

E. H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001.

[12]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[13]

S. I. Pekar, Untersuchung Über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[14]

Y. SuH. ChenS. Liu and X. Fang, Fractional Schrödinger-Poisson system with weighted Hardy potential and critical exponent, Electron. J. Differ. Equ., 2020 (2020), 1-17. 

[15]

J. T. SunT. F. Wu and Z. Feng, Non-autonomousSchrödinger-Poisson system in ${\Bbb R}^3$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077.

[16]

L. WeiX.Y. Cheng and Z. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Discrete Contin. Dyn. Syst., 36 (2016), 7169-7189.  doi: 10.3934/dcds.2016112.

[17]

J. Yang and F. Wu, Doubly critical problems involving fractional Laplacians in $\Bbb R^N$, Adv. Nonlinear Stud., 17 (2017), 677-690.  doi: 10.1515/ans-2016-6012.

[18]

R. Yang and Z. X. Lv, The properties of positive solutions to semilinear equations involving the fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 1073-1089.  doi: 10.3934/cpaa.2019052.

show all references

References:
[1]

O. BourgetM. Courdurier and C. Fernandez, Construction of solutions for some localized nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 39 (2019), 841-862.  doi: 10.3934/dcds.2019035.

[2]

L. Caffarelli, Non-local diffusions, drifts and games, pp. 37-52 in "Nonlinear Partial Differential Equations" edt by H. Holden and K. Karlsen, Abel Symp., vol. 7, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25361-4.

[3]

A. Cotsiolis and N. K. Travoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[4]

P. d'AveniaG. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[5]

S. Dipierro, L. Montoro, I. Peral, B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations, 55 (2016), Art. 99. doi: 10.1007/s00526-016-1032-5.

[6]

M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.

[7]

T. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal., 268 (2015), 3278-3289.  doi: 10.1016/j.jfa.2015.03.016.

[8]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176.  doi: 10.1016/j.jfa.2013.08.027.

[9]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[10]

E. H. Lieb and H. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174. 

[11]

E. H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001.

[12]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[13]

S. I. Pekar, Untersuchung Über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[14]

Y. SuH. ChenS. Liu and X. Fang, Fractional Schrödinger-Poisson system with weighted Hardy potential and critical exponent, Electron. J. Differ. Equ., 2020 (2020), 1-17. 

[15]

J. T. SunT. F. Wu and Z. Feng, Non-autonomousSchrödinger-Poisson system in ${\Bbb R}^3$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077.

[16]

L. WeiX.Y. Cheng and Z. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Discrete Contin. Dyn. Syst., 36 (2016), 7169-7189.  doi: 10.3934/dcds.2016112.

[17]

J. Yang and F. Wu, Doubly critical problems involving fractional Laplacians in $\Bbb R^N$, Adv. Nonlinear Stud., 17 (2017), 677-690.  doi: 10.1515/ans-2016-6012.

[18]

R. Yang and Z. X. Lv, The properties of positive solutions to semilinear equations involving the fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 1073-1089.  doi: 10.3934/cpaa.2019052.

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