Let $c(x,z)=|x|^2\cdot z^2$, and $b(x,z)=(1,1,\ldots,1),~U\subset \mathbb{R}^2,~\Gamma \subset \partial U$ and $u(x)=g(x)$ on $\Gamma$. If $\sum^n_{i=1}u_{x_i}+|x|^2u^2(x)=0$, then find the solutions for $z,x_1,x_2,\ldots$
Attempt:
$\sum^2_{i=1}u_{x_i}+|x|^2u^2(x)=0$ implies that $u_{x_1}+u_{x_2}+|x|^2u^2(x)=0$. Since $U\subset \mathbb{R}^2$, $u(x)$ is quasilinear equation and has the form $F(\Delta u,u,x)=b(x,u(x))\cdot\Delta u(x)=c(x,u(x))=0$. I know I must obtain some characteristic equations for the quasilinear PDE:
(i) $\dot{x}(s)=b(x(s),z(s))$
(ii) $\dot{z}(s)=-c(x(s),z(s))$
But I am failing to succeed.