January  2019, 18(1): 285-300. doi: 10.3934/cpaa.2019015

Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  November 2017 Revised  November 2017 Published  August 2018

Fund Project: The research is supported by National Natural Science Foundation of China (No.11471267).

In this paper, we investigate the following a class of Choquard equation
$\begin{equation*} -Δ u+u = (I_α*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N,\end{equation*}$
where
$N≥ 3,~α∈ (0,N),~I_α$
is the Riesz potential and
$F(s) = ∈t_{0}^{s}f(t)dt$
. If
$f$
satisfies almost necessary the upper critical growth conditions in the spirit of Berestycki and Lions, we obtain the existence of positive radial ground state solution by using the Pohožaev manifold and the compactness lemma of Strauss.
Citation: Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
References:
[1]

C. O. AlvesF. GaoM. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009.  Google Scholar

[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.  Google Scholar

[3]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl.(9), 58 (1978), 137-151.   Google Scholar

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

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J. Chabrowski, On the existence of G-symmetric entire solutions for semilinear elliptic equations, Rend. Circ. Mat. Palermo (2), 41 (1992), 413-440.  doi: 10.1007/BF02848946.  Google Scholar

[6]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math.. doi: 10.1142/S0219199717500377.  Google Scholar

[7]

F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006-1041.  doi: 10.1016/j.jmaa.2016.11.015.  Google Scholar

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F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math. doi: 10.1007/s11425-016-9067-5.  Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[10]

K. R. W. Jones, Newtonian quantum gravity, Australian J. Phys., 48 (1995), 1055-1081.   Google Scholar

[11]

T. KüpperZ. Zhang and H. Xia, Multiple positive solutions and bifurcation for an equation related to Choquard's equation, Proc. Edinb. Math. Soc. (2), 46 (2003), 597-607.  doi: 10.1017/S0013091502000779.  Google Scholar

[12]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.   Google Scholar

[13]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[14]

E. H. Lieb and M. Loss, Analysis, 2nd edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.  Google Scholar

[15]

J. LiuJ. F. Liao and C. L. Tang, Ground state solution for a class of Schrödinger equations involving general critical growth term, Nonlinearity, 30 (2017), 899-911.  doi: 10.1088/1361-6544/aa5659.  Google Scholar

[16]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[17]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[18]

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579 doi: 10.1090/S0002-9947-2014-06289-2.  Google Scholar

[19]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[20]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 12 pp. doi: 10.1142/S0219199715500054.  Google Scholar

[21]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[22]

S. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie Verlag. Berlin. 1954. Google Scholar

[23]

R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.  doi: 10.1007/BF02105068.  Google Scholar

[24]

W. A. Struwe, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[25]

P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.  Google Scholar

[26]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

C. O. AlvesF. GaoM. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009.  Google Scholar

[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.  Google Scholar

[3]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl.(9), 58 (1978), 137-151.   Google Scholar

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[5]

J. Chabrowski, On the existence of G-symmetric entire solutions for semilinear elliptic equations, Rend. Circ. Mat. Palermo (2), 41 (1992), 413-440.  doi: 10.1007/BF02848946.  Google Scholar

[6]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math.. doi: 10.1142/S0219199717500377.  Google Scholar

[7]

F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006-1041.  doi: 10.1016/j.jmaa.2016.11.015.  Google Scholar

[8]

F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math. doi: 10.1007/s11425-016-9067-5.  Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[10]

K. R. W. Jones, Newtonian quantum gravity, Australian J. Phys., 48 (1995), 1055-1081.   Google Scholar

[11]

T. KüpperZ. Zhang and H. Xia, Multiple positive solutions and bifurcation for an equation related to Choquard's equation, Proc. Edinb. Math. Soc. (2), 46 (2003), 597-607.  doi: 10.1017/S0013091502000779.  Google Scholar

[12]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.   Google Scholar

[13]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[14]

E. H. Lieb and M. Loss, Analysis, 2nd edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.  Google Scholar

[15]

J. LiuJ. F. Liao and C. L. Tang, Ground state solution for a class of Schrödinger equations involving general critical growth term, Nonlinearity, 30 (2017), 899-911.  doi: 10.1088/1361-6544/aa5659.  Google Scholar

[16]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[17]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[18]

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579 doi: 10.1090/S0002-9947-2014-06289-2.  Google Scholar

[19]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[20]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 12 pp. doi: 10.1142/S0219199715500054.  Google Scholar

[21]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[22]

S. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie Verlag. Berlin. 1954. Google Scholar

[23]

R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.  doi: 10.1007/BF02105068.  Google Scholar

[24]

W. A. Struwe, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[25]

P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.  Google Scholar

[26]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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