Elliptic systems with nonlinear diffusion and a convection term

Lucio Boccardo; Luigi Orsina

doi:10.3934/dcds.2022056

Article Contents

2023, Volume 43, Issue 3&4: 1052-1069. Doi: 10.3934/dcds.2022056

This issue Previous Article Trace and boundary singularities of positive solutions of a class of quasilinear equations Next Article Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations

Elliptic systems with nonlinear diffusion and a convection term

Lucio Boccardo^1, , and
Luigi Orsina^2,

1.
Dipartimento di Matematica, Sapienza Università di Roma - Istituto Lombardo, Italy
2.
Dipartimento di Matematica, Sapienza Università di Roma, Italy

^*Corresponding author: Lucio Boccardo
^*Corresponding author: Lucio Boccardo

Received: October 2021

Revised: February 2022

Early access: April 2022

Published: March 2023

Abstract / Introduction Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

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])

Keywords:

Mathematics Subject Classification: Primary: 35J15, 35J47; Secondary: 35J65.

Citation:

Full Text(HTML)

Figure 1. The admissible values of $ {\lambda} $ and $ \rho $ in Theorem 4.1

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.