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, \(H(\theta )+g(x,\theta )=\sigma (\theta )|\nabla \psi |^{2}, {\text {div}}(\sigma (\theta ) \nabla \psi )=0\) $H(\theta )+g(x,\theta )={\sigma (\theta )|\mathrm{\nabla }\psi |}^2,\text{div}(\sigma (\theta )\mathrm{\nabla }\psi )=0$ in \(\Omega \) $\mathrm{\Omega }$ where \(g(x,\theta )\) $g(x,\theta )$ is a lower order term satisfies the sign condition but without any restriction on its growth and the operator H is of the form \(\begin{aligned} H (\theta )=\sum _{|\nu |=0}^{r}(-1)^{|\nu |} D^\nu \left( h_\nu \left( x, D^\gamma \theta \right) \right) , \quad |\gamma | \le |\nu |, \end{aligned}\) $$\begin{array}{c}\hfill H(\theta )=\sum _{|\nu |=0}^r{(-1)}^{|\nu |}{D}^{\nu }\left(h_{\nu },\left(x,,,D^{\gamma },\theta \right)\right),|\gamma |\le |\nu |,\end{array}$$ is proved in the framework of Sobolev space of finite order.