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High and low perturbations of Choquard equations with critical reaction and variable growth
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 |
$ - \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, $ |
$ r(\cdot) $ |
$ g(\cdot ,\cdot) $ |
$ g(\cdot ,\cdot) $ |
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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