doi: 10.3934/cpaa.2021111
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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

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

Received  December 2020 Revised  May 2021 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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

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

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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

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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.   Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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