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Thanks to the answers in other questions, I am aware of the fact that NDSolve produces

Nonlinear coefficients are not supported in this version of NDSolve

when some boundary conditions are missed or not well put. I have the following code and I am getting the same error, and I'd really appreciate if someone could help me point out what I am doing wrong.

I have a non-linear first order partial differential equation. I want a solution for the region

Ω = RegionDifference[Ball[{0, 0}, 2], Ball[{0, 0}, 1]];

I write the following

NDSolveValue[{x^2 D[u[x, y], x]^2 + y^2 D[u[x, y], y]^2 - 
3 u[x, y]^2 == -3 - 6 x^4 + 13 x^8 - 12 x^2 y^2 + 20 x^6 y^2 - 6 y^4 + 14 x^4 y^4 + 20 x^2 y^6 + 13 y^8, DirichletCondition[u[x, y] == 2, x^2 + y^2 == 1], DirichletCondition[u[x, y] == 17, x^2 + y^2 == 4]}, u, {x,y} ∈ Ω]

This is a simplified version of the equation and region I really want to study. I chose the right hand side of the differential equation (and the boundary conditions) by hand so that it corresponds to the solution

u = 1 + (x^2 + y^2)^2

I am providing the boundary conditions for both boundaries just in case the issue comes from lack of information. Could you tell me what's the mistake please?

Michael E2
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secavara
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    The problem is not the boundary condition, but that your equation is non-linear (it has $u^2$ and $(\partial u/\partial x)^2$). The current version of Mathematica (10.3) does not support solving non-linear PDEs with the finite element method. – Szabolcs Feb 03 '16 at 17:07
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    Since your region is not rectangular, most of the answer from the duplicate won't apply. I am referring specifically to this part instead: "Concerning FEM, no, currently (V10) can not deal with non-linear stuff out of the box, it will in a future version. You can, however, use the low level FEM functions to code one up your self today. Give that a shot." See also the comments under that answer. – Szabolcs Feb 03 '16 at 17:09

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