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

Youpei Zhang; Xianhua Tang; Vicenţiu D. Rădulescu

doi:10.3934/dcds.2021180

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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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021180

References:

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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	X. Li and S. Ma, Choquard equations with critical nonlinearities, Commun. Contemp. Math., 22 (2020), 1950023, 28 pp. doi: 10.1142/S0219199719500238. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210. Google Scholar
[22]	S. Pekar, Untersuchung Über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar
[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. Google Scholar
[24]	R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	M. Willem, Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, Springer, New York, 2013. doi: 10.1007/978-1-4614-7004-5. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	V. I. Bogachev, Measure Theory, volume I, Springer-Verlag, Berlin, Heidelberg, 2007. doi: 10.1007/978-3-540-34514-5. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474. doi: 10.1090/S0273-0979-1979-14595-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	X. Li and S. Ma, Choquard equations with critical nonlinearities, Commun. Contemp. Math., 22 (2020), 1950023, 28 pp. doi: 10.1142/S0219199719500238. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer-Verlag, Berlin, 1983. doi: 10.1007/BFb0072210. Google Scholar
[22]	S. Pekar, Untersuchung Über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar
[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. Google Scholar
[24]	R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	M. Willem, Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, Springer, New York, 2013. doi: 10.1007/978-1-4614-7004-5. Google Scholar
[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. Google Scholar

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

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