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 and 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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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. 

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[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.

[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.

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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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. 

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[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.

[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.

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