doi: 10.3934/dcds.2022056
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Elliptic systems with nonlinear diffusion and a convection term

1. 

Dipartimento di Matematica, Sapienza Università di Roma - Istituto Lombardo, Italy

2. 

Dipartimento di Matematica, Sapienza Università di Roma, Italy

*Corresponding author: Lucio Boccardo

Received  October 2021 Revised  February 2022 Early access April 2022

In this paper we prove existence (and summability properties) of solutions for the following elliptic system
$ \left\{ \begin{array}{cl} -{\rm{div}}(A(x)\,{\nabla} u) + u^{{\lambda}} = -{\rm{div}}( u^{{\lambda}} \, M(x)\,{\nabla}\psi) + f(x)\,, & {\rm{in }}\; \Omega , \\ -{\rm{div}}(M(x)\,{\nabla}\psi) = u^{\rho}\,, & {\rm{in }}\; \Omega , \\ u = 0 = \psi & \;{\rm{on}}\; \partial\Omega , \end{array} \right. $
under some assumptions on
$ {\lambda} > 0 $
,
$ \rho > 0 $
and
$ f(x) $
in
$ L^{{m}}(\Omega) $
,
$ m \geq 1 $
.
      "Return of the Patriarca" (see [3])
Citation: Lucio Boccardo, Luigi Orsina. Elliptic systems with nonlinear diffusion and a convection term. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022056
References:
[1]

P. BénilanL. BoccardoT. GallouëtR. GariepyM. 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., 22 (1995), 241-273. 

[2]

P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, J. Evol. Equ., 3 (2003), 673-770.  doi: 10.1007/s00028-003-0117-8.

[3]

L. Boccardo, Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data, Ann. Mat. Pura Appl., 188 (2009), 591-601.  doi: 10.1007/s10231-008-0090-5.

[4]

L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations, 258 (2015), 2290-2314.  doi: 10.1016/j.jde.2014.12.009.

[5]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.  doi: 10.1080/03605309208820857.

[6]

L. BoccardoT. Gallouët and J. L. Vázquez, Nonlinear elliptic equations in ${\bf{R}}^N$ without growth restrictions on the data, J. Differential Equations, 105 (1993), 334-363.  doi: 10.1006/jdeq.1993.1092.

[7]

L. Boccardo and L. Orsina, Sublinear elliptic systems with a convection term, Comm. Partial Differential Equations, 45 (2020), 690-713.  doi: 10.1080/03605302.2020.1712417.

[8]

H. Brézis and F. E. Browder, Strongly nonlinear elliptic boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 587-603. 

[9]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[10]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.  doi: 10.5802/aif.204.

[11] J. L. Vázquez, The Porous Medium Equation, Oxford University Press, Oxford, 2007. 

show all references

References:
[1]

P. BénilanL. BoccardoT. GallouëtR. GariepyM. 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., 22 (1995), 241-273. 

[2]

P. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, J. Evol. Equ., 3 (2003), 673-770.  doi: 10.1007/s00028-003-0117-8.

[3]

L. Boccardo, Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data, Ann. Mat. Pura Appl., 188 (2009), 591-601.  doi: 10.1007/s10231-008-0090-5.

[4]

L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations, 258 (2015), 2290-2314.  doi: 10.1016/j.jde.2014.12.009.

[5]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.  doi: 10.1080/03605309208820857.

[6]

L. BoccardoT. Gallouët and J. L. Vázquez, Nonlinear elliptic equations in ${\bf{R}}^N$ without growth restrictions on the data, J. Differential Equations, 105 (1993), 334-363.  doi: 10.1006/jdeq.1993.1092.

[7]

L. Boccardo and L. Orsina, Sublinear elliptic systems with a convection term, Comm. Partial Differential Equations, 45 (2020), 690-713.  doi: 10.1080/03605302.2020.1712417.

[8]

H. Brézis and F. E. Browder, Strongly nonlinear elliptic boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 587-603. 

[9]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[10]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.  doi: 10.5802/aif.204.

[11] J. L. Vázquez, The Porous Medium Equation, Oxford University Press, Oxford, 2007. 
Figure 1.  The admissible values of $ {\lambda} $ and $ \rho $ in Theorem 4.1
Figure 2.  Figure 2: the admissible values of $ {\lambda} $ and $ \rho $ in Theorem 4.2 (dark gray area)
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