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

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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.

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

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

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References

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