Poisson equation with a implicit nonlinearity

Consider the nonlinear Poisson equation for u=u(x,y) as follows:

u_xx+ u_yy = f(u) in {x,y | x^2+y^2<1}

x u_x+y u_y=0 on {x,y | x^2+y^2=1}

Here f(u) is a function which is implicit in u, and f(u) cannot have an explicit function. Moreover, by using NDSolve f(u) can be numerically plotted point by point.

Is it possible to use Mathematica to numerically solve this boundary value problem? Thank you very much!

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