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

Abstract

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) = -diva(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.