Quasilinear Degenerated Elliptic Systems with Weighted in Divergence Form with Weak Monotonicity with General Data

We consider, for a bounded open domain Ω in n IR and a function : m u IR Ω → , the quasilinear elliptic system: We generalize the system ( ) ( ) , f g QES in considering a right hand side depending on the jacobian matrix Du . Here, the star in ( ) ( ) , f g QES indicates that f may depend on Du . In the right hand side, v belongs to the dual space  , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ , but with only very mild monotonicity assump-tions.


Abstract
, f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ , but with only very mild monotonicity assumptions.

Introduction
In this paper, the main point is that we do not require monotonicity in the strict monotonicity of a typical Leray-Lions operator as it is usually assumed in previous papers. The aims of this text are to prove analogous existence results under relaxed monotonicity, in particular under strict quasi-monotonicity. The main technical tool we advocate and use throughout the proof is Young measures. By applying a Galerkin schema, we obtain easily an approximating sequence k u . The Ball theorem [1] and especially the resulting tool mode available by Hungerbühler to partial differential equation theory give them a sufficient control on the gradient approximating sequence k Du to pass to the limit. This method is used by Dolzmann [2], G. J. Minty [3], H. Brezis [4], H. E. Stromberg [5], Muller [6], J. L. Lions [7], Kristznsen, J. Lower [8], M. I. Visik [9] and mainly by Hungurbühler to get the existence of a weak solution for the quasi-linear elliptic system [10]. This paper can be seen as generalization of Hungurbühler and as a continuation of Y-Akdim [11].
This kind of problems finds its applications in the model of Thomas-Fermis in atomic physics [12], and also porous flow modeling in reservoir [13].

4) ( )
, , x u F σ is strictly p-quasi-monotone in F, i.e., The main point is that we do not require strict monotonicity or monotonicity in the variables ( ) , u F in (H 3 ) as it is usually assumed in previous work (see [15] or [16] Our aim of this paper is to prove the existence of the problem ( ) , ., : -Exploiting the convergence in measure of the gradients of the approximating solutions, we will prove the following theorem. G . In order to prove theorems, we will apply a Galerkin scheme, with this aim in view, we establish in the following subsections, the key ingredient to pass to the limit for this, we assume that the conditions:  Firstly, by virtue of the growth conditions (H 2 ) and the Hölder inequality, one has − , and thanks to Hardy inequality we have: In order to find such a solution we apply a Galerkin scheme.
which provided that the negative part of ( ) So, by combining (4.5) and (4.6), we deduce that Similarly to ( ) 1, , 1, , 1 1, , 1, , By the equivalence of the norm in ( ) For all 0 k k ≥ , which implies that   On the other hand (4.9), integrating over Ω , and using the div-cul inequality we have:  L Ω , and the weakly limit of g is given by