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April  2022, 42(4): 1971-2003. doi: 10.3934/dcds.2021180

High and low perturbations of Choquard equations with critical reaction and variable growth

Youpei Zhang 1,2, , Xianhua Tang 1, and  Vicenţiu D. Rădulescu 2,3,,

1. 

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China
2. 

Department of Mathematics, University of Craiova, 200585 Craiova, Romania
3. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland

*Corresponding author: Vicenţiu D. Rădulescu

Received  June 2021 Published  April 2022 Early access  November 2021

We are concerned with the existence of ground state solutions to the nonhomogeneous perturbed Choquard equation
$ - \Delta_{p(x)} u + V(x)|u|^{p(x) - 2} u $
$ = \left( \int_{\mathbb R^N} r(y)^{-1}|u(y)|^{r(y)}|x-y|^{-\lambda(x,y)} dy\right) |u|^{r(x)-2} u+g(x,u)\ \mbox{in}\ \mathbb R^N, $
where the exponent
$ r(\cdot) $
 is critical with respect to the Hardy-Littlewood-Sobolev inequality for variable exponents. We first consider the case where the perturbation
$ g(\cdot ,\cdot) $
 is subcritical and we distinguish between the superlinear and sublinear cases. In both situations we establish the existence of solutions and we prove the asymptotic behavior of low-energy solutions in the case of high perturbations. Next, we study the case where the nonlinearity
$ g(\cdot ,\cdot) $
 is critical. We prove the existence of solutions both for low and high perturbations and we establish asymptotic properties of low-energy solutions.
Keywords: Choquard equation, anisotropic Sobolev space, Hardy-Littlewood-Sobolev inequality, Brezis-Lieb lemma, concentration-compactness principle, variational methods, critical exponent.
Mathematics Subject Classification: Primary: 47G20; Secondary: 35B38, 58E50.
Citation: Youpei Zhang, Xianhua Tang, Vicenţiu D. Rădulescu. High and low perturbations of Choquard equations with critical reaction and variable growth. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1971-2003. doi: 10.3934/dcds.2021180
References:
[1]

C. O. Alves, Existence of radial solutions for a class of $p(x)$-Laplacian equations with critical growth, Differential Integral Equations, 23 (2010), 113-123. 

[2]

C. O. Alves and L. S. Tavares, A Hardy-Littlewood-Sobolev-type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent, Mediterr. J. Math., 16 (2019), Paper No. 55, 27 pp. doi: 10.1007/s00009-019-1316-z.

[3]

V. I. Bogachev, Measure Theory, volume I, Springer-Verlag, Berlin, Heidelberg, 2007. doi: 10.1007/978-3-540-34514-5.

[4]

S. Chen, A. Fiscella, P. Pucci and X. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268 (2020), 2672-2716.  doi: 10.1016/j.jde.2019.09.041.

[5]

S. Chen and X. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.  doi: 10.1016/j.jde.2019.08.036.

[6]

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.

[7]

M. D. Donsker and S. R. S. Varadhan, Asymptotics for the polaron, Comm. Pure Appl. Math., 36 (1983), 505-528.  doi: 10.1002/cpa.3160360408.

[8]

I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6.

[9]

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

[10]

H. Fröhlich, Theory of electrical breakdown in ionic crystal, Proc. Roy. Soc. Edinburgh Sect. A, 160 (1937), 230-241.  doi: 10.1098/rspa.1937.0106.

[11]

Y. Fu and X. Zhang, Multiple solutions for a class of p(x)-Laplacian equations involving the critical exponent, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1667-1686.  doi: 10.1098/rspa.2009.0463.

[12]

M. Ghergu and S. D. Taliaferro, Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217.  doi: 10.1016/j.jde.2016.03.004.

[13]

D. Giulini and A. Großardt, The Schrödinger-Newton equation as a non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields, Classical Quantum Gravity, 29 (2012), 215010, 25 pp. doi: 10.1088/0264-9381/29/21/215010.

[14]

K. R. W. Jones, Gravitational self-energy as the litmus of reality, Modern Physics Letters A, 10 (1995), 657-668.  doi: 10.1142/S0217732395000703.

[15]

I. H. Kim and Y.-H. Kim, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math., 147 (2015), 169-191.  doi: 10.1007/s00229-014-0718-2.

[16]

X. Li and S. Ma, Choquard equations with critical nonlinearities, Commun. Contemp. Math., 22 (2020), 1950023, 28 pp. doi: 10.1142/S0219199719500238.

[17]

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.

[18]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[19]

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.

[20]

G. Mingione and V. D. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), Paper No. 125197, 41 pp. doi: 10.1016/j.jmaa.2021.125197.

[21]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[22]

S. Pekar, Untersuchung Über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[23]

R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256.

[24]

R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005.

[25]

D. Qin, V. D. Rădulescu and X. Tang, Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations, J. Differential Equations, 275 (2021), 652-683.  doi: 10.1016/j.jde.2020.11.021.

[26] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2015.  doi: 10.1201/b18601.
[27]

F. E. Schunck and E. W. Mielke, General relativistic boson stars, Classical Quantum Gravity, 20 (2003), R301–R356. doi: 10.1088/0264-9381/20/20/201.

[28]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics, vol. 34, Springer-Verlag, Berlin, 2008.

[29]

X. Tang, S. Chen, X. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268 (2020), 4663-4690.  doi: 10.1016/j.jde.2019.10.041.

[30]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.

[31]

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

[32]

M. Willem, Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, Springer, New York, 2013. doi: 10.1007/978-1-4614-7004-5.

[33]

J. Xia and Z.-Q. Wang, Saddle solutions for the Choquard equation, Calc. Var. Partial Differential Equations, 58 (2019), Art. 85, 30 pp. doi: 10.1007/s00526-019-1546-8.

show all references

References:
[1]

C. O. Alves, Existence of radial solutions for a class of $p(x)$-Laplacian equations with critical growth, Differential Integral Equations, 23 (2010), 113-123. 

[2]

C. O. Alves and L. S. Tavares, A Hardy-Littlewood-Sobolev-type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent, Mediterr. J. Math., 16 (2019), Paper No. 55, 27 pp. doi: 10.1007/s00009-019-1316-z.

[3]

V. I. Bogachev, Measure Theory, volume I, Springer-Verlag, Berlin, Heidelberg, 2007. doi: 10.1007/978-3-540-34514-5.

[4]

S. Chen, A. Fiscella, P. Pucci and X. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268 (2020), 2672-2716.  doi: 10.1016/j.jde.2019.09.041.

[5]

S. Chen and X. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.  doi: 10.1016/j.jde.2019.08.036.

[6]

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.

[7]

M. D. Donsker and S. R. S. Varadhan, Asymptotics for the polaron, Comm. Pure Appl. Math., 36 (1983), 505-528.  doi: 10.1002/cpa.3160360408.

[8]

I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6.

[9]

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

[10]

H. Fröhlich, Theory of electrical breakdown in ionic crystal, Proc. Roy. Soc. Edinburgh Sect. A, 160 (1937), 230-241.  doi: 10.1098/rspa.1937.0106.

[11]

Y. Fu and X. Zhang, Multiple solutions for a class of p(x)-Laplacian equations involving the critical exponent, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1667-1686.  doi: 10.1098/rspa.2009.0463.

[12]

M. Ghergu and S. D. Taliaferro, Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217.  doi: 10.1016/j.jde.2016.03.004.

[13]

D. Giulini and A. Großardt, The Schrödinger-Newton equation as a non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields, Classical Quantum Gravity, 29 (2012), 215010, 25 pp. doi: 10.1088/0264-9381/29/21/215010.

[14]

K. R. W. Jones, Gravitational self-energy as the litmus of reality, Modern Physics Letters A, 10 (1995), 657-668.  doi: 10.1142/S0217732395000703.

[15]

I. H. Kim and Y.-H. Kim, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math., 147 (2015), 169-191.  doi: 10.1007/s00229-014-0718-2.

[16]

X. Li and S. Ma, Choquard equations with critical nonlinearities, Commun. Contemp. Math., 22 (2020), 1950023, 28 pp. doi: 10.1142/S0219199719500238.

[17]

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.

[18]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[19]

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.

[20]

G. Mingione and V. D. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl., 501 (2021), Paper No. 125197, 41 pp. doi: 10.1016/j.jmaa.2021.125197.

[21]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[22]

S. Pekar, Untersuchung Über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[23]

R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256.

[24]

R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005.

[25]

D. Qin, V. D. Rădulescu and X. Tang, Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations, J. Differential Equations, 275 (2021), 652-683.  doi: 10.1016/j.jde.2020.11.021.

[26] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2015.  doi: 10.1201/b18601.
[27]

F. E. Schunck and E. W. Mielke, General relativistic boson stars, Classical Quantum Gravity, 20 (2003), R301–R356. doi: 10.1088/0264-9381/20/20/201.

[28]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics, vol. 34, Springer-Verlag, Berlin, 2008.

[29]

X. Tang, S. Chen, X. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268 (2020), 4663-4690.  doi: 10.1016/j.jde.2019.10.041.

[30]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.

[31]

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

[32]

M. Willem, Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, Springer, New York, 2013. doi: 10.1007/978-1-4614-7004-5.

[33]

J. Xia and Z.-Q. Wang, Saddle solutions for the Choquard equation, Calc. Var. Partial Differential Equations, 58 (2019), Art. 85, 30 pp. doi: 10.1007/s00526-019-1546-8.

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