Thanks for the insight! :)

@GiorgioMetafune I was trying that initially and couldn't use the maximum principle directly. I got $v(x) = \frac{f(0)}{2}x_n^2 - \frac{f(0)}{2}\int_{\mathbb{S}^{n-1}} K(x,y) y_n^2\operatorname{sgn}(y_n)\,d\sigma(y)$ (where, $K$ is the Poisson Kernel for unit sphere).. which is deg $2$ polynomial + $f(0) \times$ harmonic and I didn't know how to bound the $f(0)$ factor with $\lVert v \rVert_{L^{\infty}(B_1^+)}$.

Thank you very much! :) I couldn't bound $|f(0)|$ with $\lVert u_0 \rVert_{L^{\infty}(B_{1/2}^+)}$. Could you tell me how you guessed the barrier function?