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November  2019, 12(7): 2035-2050. doi: 10.3934/dcdss.2019131

Existence of positive ground state solutions for Choquard equation with variable exponent growth

Gui-Dong Li and  Chun-Lei Tang ,

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

* Corresponding author: Chun-Lei Tang (tangcl@swu.edu.cn)

Received  November 2017 Revised  May 2018 Published  December 2018

Fund Project: Supported by National Natural Science Foundation of China (No.11471267).

In this paper, we investigate the following Choquard equation
$ \begin{equation*} -\Delta u = (I_\alpha*|u|^{f(x)})|u|^{f(x)-2}u \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $
where
$ N\geq 3 $
,
$ \alpha\in (0,N) $
 and
$ I_\alpha $
 is the Riesz potential. If
$ \begin{equation*} f(x) = \begin{cases} p, &x\in\Omega,\\ ( N+\alpha)/(N-2), &x\in\mathbb{R}^N \backslash\Omega, \end{cases} \end{equation*} $
where
$ 1< p <\frac{N+\alpha}{N-2} $
 and
$ \Omega\subset\mathbb{R}^N $
 is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.
Keywords: Choquard equation, zero mass, Nehari manifold, positive ground state solutions, variable exponent growth.
Mathematics Subject Classification: Primary: 35A15, 35B50; Secondary: 35J61.
Citation: Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
References:
[1]

E. Acerbi and G. Mingione, Regularity results for stationary electrorheological fuids, Arch. Ration. Mech. Anal., 164 (2002), 213-259.  doi: 10.1007/s00205-002-0208-7.

[2]

C. O. Alves, Existence of solution for a degenerate $p(x)$-Laplacian equation in $\mathbb{R}^N$, J. Math. Anal. Appl., 345 (2008), 731-742.  doi: 10.1016/j.jmaa.2008.04.060.

[3]

C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrodinger equation in $\mathbb{R}^2$, J. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.

[4]

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

[5]

C. O. Alves, A. Nobrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9.

[6]

C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via the penalization method, Proc. Roy. Soc. Edinb. A Math., 146 (2015), 23-58.  doi: 10.1017/S0308210515000311.

[7]

S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 65 (2006), 728-761.  doi: 10.1016/j.na.2005.09.035.

[8]

S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36.  doi: 10.1007/s11565-006-0002-9.

[9]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Diferential Geom., 11 (1976), 573-598.  doi: 10.4310/jdg/1214433725.

[10]

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.

[11]

L. Calotă, On some quasilinear elliptic equations with critical Sobolev exponents and non-standard growth conditions, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 249-256. 

[12]

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.  doi: 10.1007/s002110050258.

[13]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20 (2018), 1750037, 22 pp. doi: 10.1142/S0219199717500377.

[14]

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.

[15]

F. Gao and M. Yang, On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5.

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977.

[17]

G.-D. Li and C.-L. Tang, Existence of ground state solutions for Choquard equation involving the general upper critical, Commun. Pure Appl. Anal. 18 (2019) 285-300. doi: 10.3934/cpaa.2019015.

[18]

G.-D. Li and C.-L. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl. 76 (2018), 2635-2647.

[19]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.

[20]

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.

[21]

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

[22]

J. Liu, J.-F. Liao and C.-L. Tang, Ground state solutions for semilinear elliptic equations with zero mass in $\mathbb{R}^N$, Electron. J. Differential Equations, 2015 (2015), 1-11. 

[23]

D. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48.  doi: 10.1016/j.na.2013.12.022.

[24]

D. Lü, Existence and Concentration of Solutions for a Nonlinear Choquard Equation, Mediterr. J. Math., 12 (2015), 839-850.  doi: 10.1007/s00009-014-0428-8.

[25]

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.

[26]

R. A. Mashiyev, B. Cekic, M. Avci and Z. Yucedag, Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Var. Elliptic Equ., 57 (2012), 579-595.  doi: 10.1080/17476933.2011.598928.

[27]

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.

[28]

I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997). Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[29]

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.

[30]

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.

[31]

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.

[32]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp. doi: 10.1142/S0219199715500054.

[33]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x.

[34]

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

[35]

P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.  doi: 10.1016/j.jde.2014.05.023.

[36]

M. Råužička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[37]

M. Riesz, L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1-223.  doi: 10.1007/BF02395016.

[38]

D. Ruiz and J. Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations, 264 (2018), 1231-1262.  doi: 10.1016/j.jde.2017.09.034.

[39]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[40]

J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 1, 24 pp. doi: 10.1007/s00030-016-0424-8.

[41]

T. Wang and T. S. Yi, Uniqueness of positive solutions of the Choquard type equations, Appl. Anal., 96 (2017), 409-417.  doi: 10.1080/00036811.2016.1138473.

[42]

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.

[43]

H. Zhang, J. Xu and F. Zhang, Bound and ground states for a concave-convex generalized Choquard equation, Acta Appl. Math. 147 (2017), 81-93. doi: 10.1007/s10440-016-0069-y.

[44]

H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814.  doi: 10.1016/j.camwa.2017.02.026.

[45]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710,877.  doi: 10.1070/IM1987v029n01ABEH000958.

show all references

References:
[1]

E. Acerbi and G. Mingione, Regularity results for stationary electrorheological fuids, Arch. Ration. Mech. Anal., 164 (2002), 213-259.  doi: 10.1007/s00205-002-0208-7.

[2]

C. O. Alves, Existence of solution for a degenerate $p(x)$-Laplacian equation in $\mathbb{R}^N$, J. Math. Anal. Appl., 345 (2008), 731-742.  doi: 10.1016/j.jmaa.2008.04.060.

[3]

C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrodinger equation in $\mathbb{R}^2$, J. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.

[4]

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

[5]

C. O. Alves, A. Nobrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9.

[6]

C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via the penalization method, Proc. Roy. Soc. Edinb. A Math., 146 (2015), 23-58.  doi: 10.1017/S0308210515000311.

[7]

S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 65 (2006), 728-761.  doi: 10.1016/j.na.2005.09.035.

[8]

S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36.  doi: 10.1007/s11565-006-0002-9.

[9]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Diferential Geom., 11 (1976), 573-598.  doi: 10.4310/jdg/1214433725.

[10]

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.

[11]

L. Calotă, On some quasilinear elliptic equations with critical Sobolev exponents and non-standard growth conditions, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 249-256. 

[12]

A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.  doi: 10.1007/s002110050258.

[13]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20 (2018), 1750037, 22 pp. doi: 10.1142/S0219199717500377.

[14]

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.

[15]

F. Gao and M. Yang, On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5.

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977.

[17]

G.-D. Li and C.-L. Tang, Existence of ground state solutions for Choquard equation involving the general upper critical, Commun. Pure Appl. Anal. 18 (2019) 285-300. doi: 10.3934/cpaa.2019015.

[18]

G.-D. Li and C.-L. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl. 76 (2018), 2635-2647.

[19]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.

[20]

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.

[21]

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

[22]

J. Liu, J.-F. Liao and C.-L. Tang, Ground state solutions for semilinear elliptic equations with zero mass in $\mathbb{R}^N$, Electron. J. Differential Equations, 2015 (2015), 1-11. 

[23]

D. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48.  doi: 10.1016/j.na.2013.12.022.

[24]

D. Lü, Existence and Concentration of Solutions for a Nonlinear Choquard Equation, Mediterr. J. Math., 12 (2015), 839-850.  doi: 10.1007/s00009-014-0428-8.

[25]

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.

[26]

R. A. Mashiyev, B. Cekic, M. Avci and Z. Yucedag, Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Var. Elliptic Equ., 57 (2012), 579-595.  doi: 10.1080/17476933.2011.598928.

[27]

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.

[28]

I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997). Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[29]

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.

[30]

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.

[31]

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.

[32]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp. doi: 10.1142/S0219199715500054.

[33]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x.

[34]

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

[35]

P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.  doi: 10.1016/j.jde.2014.05.023.

[36]

M. Råužička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[37]

M. Riesz, L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1-223.  doi: 10.1007/BF02395016.

[38]

D. Ruiz and J. Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations, 264 (2018), 1231-1262.  doi: 10.1016/j.jde.2017.09.034.

[39]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[40]

J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 1, 24 pp. doi: 10.1007/s00030-016-0424-8.

[41]

T. Wang and T. S. Yi, Uniqueness of positive solutions of the Choquard type equations, Appl. Anal., 96 (2017), 409-417.  doi: 10.1080/00036811.2016.1138473.

[42]

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.

[43]

H. Zhang, J. Xu and F. Zhang, Bound and ground states for a concave-convex generalized Choquard equation, Acta Appl. Math. 147 (2017), 81-93. doi: 10.1007/s10440-016-0069-y.

[44]

H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814.  doi: 10.1016/j.camwa.2017.02.026.

[45]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710,877.  doi: 10.1070/IM1987v029n01ABEH000958.

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