Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system

In this paper, we analyze the following nonlinear elliptic problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>ρ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>∇</m:mi> <m:mi>φ</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mtext> in </m:mtext> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mtext>div</m:mtext> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>∇</m:mi> <m:mi>φ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mtext> in </m:mtext> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>φ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>φ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>.</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> ${cases}A(u)=ρ (u) ∇ φ { }²\,in\,{Ω}, \\ div(ρ (u)∇ φ )=0\,in\,{Ω}, \\ u=0\,on\,∂ {Ω}, \\ φ ={φ }₀\,on\,∂ {Ω}. {cases}$ where A ( u ) = −div a ( x , u , ∇ u ) is a Leray-Lions operator of order p . The second member of the first equation is only in L 1 (Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.