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 A ( u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω . $\begin{cases}A\left(u\right)=\rho \left(u\right)\vert \nabla \varphi {\vert }^{2}\,\text{in}\,{\Omega},\quad \hfill \\ \text{div}\left(\rho \left(u\right)\nabla \varphi \right)=0\,\text{in}\,{\Omega},\quad \hfill \\ u=0\,\text{on}\,\partial {\Omega},\quad \hfill \\ \varphi ={\varphi }_{0}\,\text{on}\,\partial {\Omega}.\quad \hfill \end{cases}$ 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.