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July  2022, 42(7): 3301-3328. doi: 10.3934/dcds.2022016

Existence of positive solutions for a class of fractional Choquard equation in exterior domain

Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Perú

*Corresponding author: César E. Torres Ledesma

Received  June 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

Fund Project: César T. Ledesma was partially supported by CONCYTEC, Peru, 379-2019-FONDECYT "ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES"

In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and Choquard type nonlinearity. We prove the main results using variational method combined with Brouwer theory of degree and Deformation Lemma..

Citation: César E. Torres Ledesma. Existence of positive solutions for a class of fractional Choquard equation in exterior domain. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3301-3328. doi: 10.3934/dcds.2022016
References:
[1]

C. O. Alves and L. R. de Freitas, Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth, Milan J. Math., 85 (2017), 309-330.  doi: 10.1007/s00032-017-0274-9.

[2]

C. O. AlvesG. Molica Bisci and C. E. Torres Ledesma, Existence of solutions for a class of fractional elliptic problems on exterior domains, J. Differential Equations, 268 (2020), 7183-7219.  doi: 10.1016/j.jde.2019.11.068.

[3]

C. O. Alves and C. E. Torres Ledesma, Fractional elliptic problem in exterior domains with nonlocal Neumann condition, Nonlinear Analysis, 195 (2020), 111732.  doi: 10.1016/j.na.2019.111732.

[4]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[5] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing Switzerland, 2016. 
[6]

P. Chen and X. Liu, Positive solutions for a Choquard equation in exterior domain, Comm. Pure and Applied Analysis, 20 (2021), 2237-2256.  doi: 10.3934/cpaa.2021065.

[7]

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.  doi: 10.1016/j.jmaa.2013.04.081.

[8]

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

[9]

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.

[10]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth n in the Whole of $ \mathbb R^n$, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 15. Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8.

[11]

S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.

[12]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[13]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[14]

P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Differential Equations, 54 (2015), 75-98. 

[15]

R. Frank and E. Lenzmann, On ground states for the $L^2$-critical boson star equation, arXiv: 0910.2721.

[16]

M. Ghimenti and D. Pagliardini, Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains, Calc. Var. Partial Differ. Equ., 58 (2019), Paper No. 167, 21 pp. doi: 10.1007/s00526-019-1605-1.

[17]

D. Goel, The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent, Topol. Methods Nonlinear Anal., 54 (2019), 751-771.  doi: 10.12775/tmna.2019.068.

[18]

E. H. Lieb and M. P. Loss, Analysis, Amer. Math. Soc., Providence RI 2001. doi: 10.1090/gsm/014.

[19] G. Molica BisciV. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[20]

V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of super-solutions to Choquard equations in exterior domains, J. Differ. Equ., 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.

[21] C. Pozrikidis, The Fractional Laplacian, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19666.
[22]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.

[23]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24 Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

C. O. Alves and L. R. de Freitas, Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth, Milan J. Math., 85 (2017), 309-330.  doi: 10.1007/s00032-017-0274-9.

[2]

C. O. AlvesG. Molica Bisci and C. E. Torres Ledesma, Existence of solutions for a class of fractional elliptic problems on exterior domains, J. Differential Equations, 268 (2020), 7183-7219.  doi: 10.1016/j.jde.2019.11.068.

[3]

C. O. Alves and C. E. Torres Ledesma, Fractional elliptic problem in exterior domains with nonlocal Neumann condition, Nonlinear Analysis, 195 (2020), 111732.  doi: 10.1016/j.na.2019.111732.

[4]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[5] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing Switzerland, 2016. 
[6]

P. Chen and X. Liu, Positive solutions for a Choquard equation in exterior domain, Comm. Pure and Applied Analysis, 20 (2021), 2237-2256.  doi: 10.3934/cpaa.2021065.

[7]

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.  doi: 10.1016/j.jmaa.2013.04.081.

[8]

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

[9]

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.

[10]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth n in the Whole of $ \mathbb R^n$, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 15. Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8.

[11]

S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.

[12]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[13]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[14]

P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var. Partial Differential Equations, 54 (2015), 75-98. 

[15]

R. Frank and E. Lenzmann, On ground states for the $L^2$-critical boson star equation, arXiv: 0910.2721.

[16]

M. Ghimenti and D. Pagliardini, Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains, Calc. Var. Partial Differ. Equ., 58 (2019), Paper No. 167, 21 pp. doi: 10.1007/s00526-019-1605-1.

[17]

D. Goel, The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent, Topol. Methods Nonlinear Anal., 54 (2019), 751-771.  doi: 10.12775/tmna.2019.068.

[18]

E. H. Lieb and M. P. Loss, Analysis, Amer. Math. Soc., Providence RI 2001. doi: 10.1090/gsm/014.

[19] G. Molica BisciV. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[20]

V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of super-solutions to Choquard equations in exterior domains, J. Differ. Equ., 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.

[21] C. Pozrikidis, The Fractional Laplacian, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19666.
[22]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.

[23]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24 Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

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