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On the critical Schrödinger-Poisson system with $ p $-Laplacian
The duality method for mean field games systems
1. | Istituto Lombardo - Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy |
2. | Dipartimento di Matematica, Sapienza Università di Roma, P.le A. Moro 2, 00185, Roma, Italy |
$ \left\{ \begin{array}{cl} -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla} u) + u - {\mathop{{{\rm{div}}}}}(u\,A(x)\,{\nabla}\psi) = f(x) & {\rm{in \; \Omega ,}}\\ -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla}\psi) + \psi + A(x)\,{\nabla}\psi \cdot {\nabla} \psi = u^{p-1} & {\rm{in \; \Omega ,}} \\ u = 0 = \psi & {\rm{on \; \partial\Omega ,}} \end{array} \right. $ |
$ p > 1 $ |
$ f(x) $ |
References:
[1] |
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez,
An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.
|
[2] |
L. Boccardo,
Some developments on Dirichlet problems with discontinuous coefficients, Boll. Unione Mat. Ital. (9), 2 (2009), 285-297.
|
[3] |
L. Boccardo,
Dirichlet problems with singular convection terms and applications, J. Differ. Equ., 258 (2015), 2290-2314.
doi: 10.1016/j.jde.2014.12.009. |
[4] |
L. Boccardo,
Weak maximum principle for Dirichlet problems with convection or drift terms, Math. Eng., 3 (2021), 1-9.
doi: 10.3934/mine.2021026. |
[5] |
L. Boccardo and T. Gallouët,
Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., 17 (1992), 641-655.
doi: 10.1080/03605309208820857. |
[6] |
L. Boccardo and T. Gallouët,
Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579.
doi: 10.1016/0362-546X(92)90022-7. |
[7] |
L. Boccardo and T. Gallouët,
$W_0^{1, 1}$ solutions in some borderline cases of Calderon-Zygmund theory, J. Differ. Equ., 253 (2012), 2698-2714.
doi: 10.1016/j.jde.2012.07.003. |
[8] |
L. Boccardo and G. Croce, Elliptic partial differential equations, in De Gruyter Studies in Mathematics, De Gruyter, Berlin, 2014. |
[9] |
L. Boccardo, G. Croce and L. Orsina,
Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439.
doi: 10.1007/s00229-011-0473-6. |
[10] |
L. Boccardo and L. Orsina,
Strong maximum principle for some quasilinear Dirichlet problems having natural growth terms, Adv. Nonlinear Stud., 20 (2020), 503-510.
doi: 10.1515/ans-2020-2088. |
[11] |
L. Boccardo, L. Orsina and A. Porretta,
Strongly coupled elliptic equations related to mean-field games systems, J. Differ. Equ., 261 (2016), 1796-1834.
doi: 10.1016/j.jde.2016.04.018. |
[12] |
H. Brezis and A.C. Ponce,
Remarks on the strong maximum principle, Differ. Integral Equ., 16 (2003), 1-12.
|
[13] |
M. Cirant,
Stationary focusing mean-field games, Commun. Partial Differ. Equ., 41 (2016), 1324-1346.
doi: 10.1080/03605302.2016.1192647. |
[14] |
M. Cirant and A. Goffi, Maximal $L^q$-regularity for parabolic Hamilton-Jacobi equations and applications to mean field games, Ann. Partial Differ Equ., 7 (2021), 40 pp.
doi: 10.1007/s40818-021-00109-y. |
[15] |
D. A. Gomes, E. A. Pimentel and H. Sánchez-Morgado,
Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var., 22 (2016), 562-580.
doi: 10.1051/cocv/2015029. |
[16] |
D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics, Springer, 2016.
doi: 10.1007/978-3-319-38934-9. |
[17] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅰ. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019. |
[18] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅱ. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
[19] |
J. M. Lasry and P. L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[20] |
L. Orsina and A.C. Ponce,
Strong maximum principle for Schrödinger operators with singular potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 477-493.
doi: 10.1016/j.anihpc.2014.11.004. |
[21] |
G. Stampacchia,
Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.
|
show all references
References:
[1] |
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez,
An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.
|
[2] |
L. Boccardo,
Some developments on Dirichlet problems with discontinuous coefficients, Boll. Unione Mat. Ital. (9), 2 (2009), 285-297.
|
[3] |
L. Boccardo,
Dirichlet problems with singular convection terms and applications, J. Differ. Equ., 258 (2015), 2290-2314.
doi: 10.1016/j.jde.2014.12.009. |
[4] |
L. Boccardo,
Weak maximum principle for Dirichlet problems with convection or drift terms, Math. Eng., 3 (2021), 1-9.
doi: 10.3934/mine.2021026. |
[5] |
L. Boccardo and T. Gallouët,
Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., 17 (1992), 641-655.
doi: 10.1080/03605309208820857. |
[6] |
L. Boccardo and T. Gallouët,
Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579.
doi: 10.1016/0362-546X(92)90022-7. |
[7] |
L. Boccardo and T. Gallouët,
$W_0^{1, 1}$ solutions in some borderline cases of Calderon-Zygmund theory, J. Differ. Equ., 253 (2012), 2698-2714.
doi: 10.1016/j.jde.2012.07.003. |
[8] |
L. Boccardo and G. Croce, Elliptic partial differential equations, in De Gruyter Studies in Mathematics, De Gruyter, Berlin, 2014. |
[9] |
L. Boccardo, G. Croce and L. Orsina,
Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439.
doi: 10.1007/s00229-011-0473-6. |
[10] |
L. Boccardo and L. Orsina,
Strong maximum principle for some quasilinear Dirichlet problems having natural growth terms, Adv. Nonlinear Stud., 20 (2020), 503-510.
doi: 10.1515/ans-2020-2088. |
[11] |
L. Boccardo, L. Orsina and A. Porretta,
Strongly coupled elliptic equations related to mean-field games systems, J. Differ. Equ., 261 (2016), 1796-1834.
doi: 10.1016/j.jde.2016.04.018. |
[12] |
H. Brezis and A.C. Ponce,
Remarks on the strong maximum principle, Differ. Integral Equ., 16 (2003), 1-12.
|
[13] |
M. Cirant,
Stationary focusing mean-field games, Commun. Partial Differ. Equ., 41 (2016), 1324-1346.
doi: 10.1080/03605302.2016.1192647. |
[14] |
M. Cirant and A. Goffi, Maximal $L^q$-regularity for parabolic Hamilton-Jacobi equations and applications to mean field games, Ann. Partial Differ Equ., 7 (2021), 40 pp.
doi: 10.1007/s40818-021-00109-y. |
[15] |
D. A. Gomes, E. A. Pimentel and H. Sánchez-Morgado,
Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var., 22 (2016), 562-580.
doi: 10.1051/cocv/2015029. |
[16] |
D. A. Gomes, E. A. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics, Springer, 2016.
doi: 10.1007/978-3-319-38934-9. |
[17] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅰ. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019. |
[18] |
J. M. Lasry and P. L. Lions,
Jeux à champ moyen. Ⅱ. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
[19] |
J. M. Lasry and P. L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[20] |
L. Orsina and A.C. Ponce,
Strong maximum principle for Schrödinger operators with singular potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 477-493.
doi: 10.1016/j.anihpc.2014.11.004. |
[21] |
G. Stampacchia,
Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.
|
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