# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022056
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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)
 [1] Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11 [2] Li Ma, Chong Li, Lin Zhao. Monotone solutions to a class of elliptic and diffusion equations. Communications on Pure and Applied Analysis, 2007, 6 (1) : 237-246. doi: 10.3934/cpaa.2007.6.237 [3] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [4] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272 [5] Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081 [6] Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901 [7] Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i [8] Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309 [9] Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707 [10] Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713 [11] Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019 [12] Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067 [13] Y. Efendiev, Alexander Pankov. Meyers type estimates for approximate solutions of nonlinear elliptic equations and their applications. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 481-492. doi: 10.3934/dcdsb.2006.6.481 [14] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2469-2494. doi: 10.3934/dcds.2013.33.2469 [15] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [16] Song Peng, Aliang Xia. Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1201-1217. doi: 10.3934/cpaa.2018058 [17] Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331 [18] Junichi Harada, Mitsuharu Ôtani. $H^2$-solutions for some elliptic equations with nonlinear boundary conditions. Conference Publications, 2009, 2009 (Special) : 333-339. doi: 10.3934/proc.2009.2009.333 [19] Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393 [20] Dumitru Motreanu. Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 831-843. doi: 10.3934/dcdss.2012.5.831

2021 Impact Factor: 1.588

## Metrics

• HTML views (48)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar