# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022112
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## Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

 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

Received  February 2022 Revised  April 2022 Early access May 2022

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

In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence of radial ground state solutions for Choquard equations by using the refined Sobolev inequality with the Morrey norm, and show that any nonnegative weak solutions of Choquard equations have additional decay properties in terms of ground state representation.

Citation: Yu Su, Zhaosheng Feng. Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022112
##### References:
 [1] M. Ali Husaini and C. Liu, Synchronized and ground-state solutions to a coupled Schrödinger system, Commun. Pure Appl. Anal., 21 (2022), 639-667.  doi: 10.3934/cpaa.2021192. [2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [3] 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: 10.1090/S0002-9939-1983-0699419-3. [4] C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684. [5] C. Cortázar and M. García-Huidobro, On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian, Commun. Pure Appl. Anal., 5 (2006), 813-826.  doi: 10.3934/cpaa.2006.5.813. [6] V. Felli, On the existence of ground state solutions to nonlinear Schrödinger equations with multisingular inverse-square anisotropic potentials, J. Anal. Math., 108 (2009), 189-217.  doi: 10.1007/s11854-009-0023-2. [7] V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316.  doi: 10.1016/j.jfa.2006.10.019. [8] V. Felli, E. M. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete Contin. Dyn. Syst., 21 (2008), 91-119.  doi: 10.3934/dcds.2008.21.91. [9] V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multisingular inversesquare anisotropic potentials, Indiana Univ. Math. J., 58 (2009), 617-676.  doi: 10.1512/iumj.2009.58.3471. [10] V. Felli and S. Terracini, Nonlinear Schrödinger equations with symmetric multi-polar potentials, Calc. Var. Partial Differential Equations, 27 (2006), 25-58.  doi: 10.1007/s00526-006-0020-6. [11] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495.  doi: 10.1080/03605300500394439. [12] Z. Feng and Y. Su, Ground state solution to the biharmonic equation, Z. Angew. Math. Phys., 73 (2022), Paper No. 15, 24 pp. doi: 10.1007/s00033-021-01643-2. [13] E. H. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14., American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [14] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [15] G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.  doi: 10.1017/S0308210500012191. [16] W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056. [17] 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. [18] S. Terracini, On positive entire solutions to a class of equations with a singular cofficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.

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##### References:
 [1] M. Ali Husaini and C. Liu, Synchronized and ground-state solutions to a coupled Schrödinger system, Commun. Pure Appl. Anal., 21 (2022), 639-667.  doi: 10.3934/cpaa.2021192. [2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [3] 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: 10.1090/S0002-9939-1983-0699419-3. [4] C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684. [5] C. Cortázar and M. García-Huidobro, On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian, Commun. Pure Appl. Anal., 5 (2006), 813-826.  doi: 10.3934/cpaa.2006.5.813. [6] V. Felli, On the existence of ground state solutions to nonlinear Schrödinger equations with multisingular inverse-square anisotropic potentials, J. Anal. Math., 108 (2009), 189-217.  doi: 10.1007/s11854-009-0023-2. [7] V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316.  doi: 10.1016/j.jfa.2006.10.019. [8] V. Felli, E. M. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete Contin. Dyn. Syst., 21 (2008), 91-119.  doi: 10.3934/dcds.2008.21.91. [9] V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multisingular inversesquare anisotropic potentials, Indiana Univ. Math. J., 58 (2009), 617-676.  doi: 10.1512/iumj.2009.58.3471. [10] V. Felli and S. Terracini, Nonlinear Schrödinger equations with symmetric multi-polar potentials, Calc. Var. Partial Differential Equations, 27 (2006), 25-58.  doi: 10.1007/s00526-006-0020-6. [11] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495.  doi: 10.1080/03605300500394439. [12] Z. Feng and Y. Su, Ground state solution to the biharmonic equation, Z. Angew. Math. Phys., 73 (2022), Paper No. 15, 24 pp. doi: 10.1007/s00033-021-01643-2. [13] E. H. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14., American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [14] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [15] G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.  doi: 10.1017/S0308210500012191. [16] W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056. [17] 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. [18] S. Terracini, On positive entire solutions to a class of equations with a singular cofficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.
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