Existence of solutions to a strongly nonlinear elliptic coupled system of finite order

The existence of a capacity solution to the strongly nonlinear degenerate problem, namely, \(H(\theta )+g(x,\theta )=\sigma (\theta )|\nabla \psi |^{2}, {\text {div}}(\sigma (\theta ) \nabla \psi )=0\) in \(\Omega \) where \(g(x,\theta )\) is a lower order term satisfies the sign condition but without any restriction on its growth and the operator H is of the form

$$\begin{aligned} H (\theta )=\sum _{|\nu |=0}^{r}(-1)^{|\nu |} D^\nu \left( h_\nu \left( x, D^\gamma \theta \right) \right) , \quad |\gamma | \le |\nu |, \end{aligned}$$

is proved in the framework of Sobolev space of finite order.