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Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity
Generalized Schrödinger-Poisson type systems
1. | Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via dell'Ateneo Lucano 10, I-85100 Potenza, Italy |
2. | Dipartimento di Matematica, Politecnico di Bari, Via Orabona, 4, I-70125 Bari |
3. | Dipartimento di Matematica, Universita degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy |
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
A. Azzollini and P. d'Avenia, On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438.
doi: 10.1016/j.jmaa.2011.09.012. |
[4] |
A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[5] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. |
[6] |
M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328.
doi: 10.1007/s00220-003-0972-8. |
[7] |
T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. |
[8] |
P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009), e1985-e1995.
doi: 10.1016/j.na.2009.02.111. |
[9] |
P. d'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149.
doi: 10.3934/dcds.2010.26.135. |
[10] |
L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. |
[11] |
H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437. |
[12] |
E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93.
|
[13] |
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[14] |
D. Mugnai, The Schrödinger-Poisson system with positive potential, Comm. Partial Differential Equations, 36 (2011), 1099-1117.
doi: 10.1080/03605302.2011.558551. |
[15] |
L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system, Topol. Methods Nonlinear Anal., 29 (2007), 251-264. |
[16] |
L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain, Appl. Math. Lett., 21 (2008), 521-528.
doi: 10.1016/j.aml.2007.06.005. |
[17] |
S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\l f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. |
[18] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[19] |
D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. |
[20] |
G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299.
doi: 10.1016/j.jmaa.2009.10.061. |
[21] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' 4th edition, Springer-Verlag, Berlin, 2008. |
show all references
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
A. Azzollini and P. d'Avenia, On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438.
doi: 10.1016/j.jmaa.2011.09.012. |
[4] |
A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[5] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. |
[6] |
M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328.
doi: 10.1007/s00220-003-0972-8. |
[7] |
T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. |
[8] |
P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009), e1985-e1995.
doi: 10.1016/j.na.2009.02.111. |
[9] |
P. d'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149.
doi: 10.3934/dcds.2010.26.135. |
[10] |
L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. |
[11] |
H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437. |
[12] |
E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93.
|
[13] |
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[14] |
D. Mugnai, The Schrödinger-Poisson system with positive potential, Comm. Partial Differential Equations, 36 (2011), 1099-1117.
doi: 10.1080/03605302.2011.558551. |
[15] |
L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system, Topol. Methods Nonlinear Anal., 29 (2007), 251-264. |
[16] |
L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain, Appl. Math. Lett., 21 (2008), 521-528.
doi: 10.1016/j.aml.2007.06.005. |
[17] |
S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\l f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. |
[18] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[19] |
D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. |
[20] |
G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299.
doi: 10.1016/j.jmaa.2009.10.061. |
[21] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' 4th edition, Springer-Verlag, Berlin, 2008. |
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