# American Institute of Mathematical Sciences

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. 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., 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. Boccardo, T. 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. 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., 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. Boccardo, T. 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.
The admissible values of ${\lambda}$ and $\rho$ in Theorem 4.1
Figure 2: the admissible values of ${\lambda}$ and $\rho$ in Theorem 4.2 (dark gray area)
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