Non linear PDE solution

Could someone advise if it is possible to solve the following PDE with Mathematica? I am quite a beginner in Mathematica so any input would be highly appreciated.

$displaystylefrac{1}{2} sigma^2frac{partial^2 u(x,y)}{partial x^2}+frac{1}{2} sigma^2frac{partial^2 u(x,y)}{partial y^2}+afrac{partial u(x,y)}{partial x}left(frac{frac{partial u(x,y)}{partial x}+frac{partial u(x,y)}{partial y}}{3frac{partial u(x,y)}{partial x}+frac{partial u(x,y)}{partial y}}right)^2-r u(x,y)=0.$,
where $sigma$, a and r are come constants.

The domain for $sigma=0.85$, $r=0.05$ and $a=50$ is specified as follows
$y leq 0.52 + 0.46 x; x leq 0.52 + 0.46 y; xgeq 0; y geq 0; x leq 0.97; y leq 0.97$

witht the boundary conditions are

1. $u(x,y)=0$ for $x=0,yleq0.46$;
2. $u(x,y)=249.5 mathrm{e}^{-34.5784 x} (-1 + mathrm{e}^{34.6 x})$ for $y=0, xleq0.46$;
3. $frac{partial u(x,y)}{partial x}=1$ for $0.46leq xleq 0.97$, on $x = 0.52 + 0.46 y $;
4. $frac{partial u(x,y)}{partial y}=0$ for $0.46leq yleq 0.97$, on $y = 0.52 + 0.46 x $.