In a $B_R(0) \subset \mathbb{R}^2$, I have a equation $\Delta u=f(u)$ in $B_R(0)$, and $u(|x|=R)=1$. It is supposed that $0\leq u \leq 1$ and of $W^{1,2}$.$f$ is nice enough.It is clear that we have the holder continuity by some sobolev embedding.

But I want to show the holder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be test function $\phi=0$ outside the ball of radiu $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get
$$\int_{B_2r(0)}\phi^2|\nabla u|^2 +\int_{B_2r(0)}2\phi u\nabla u\nabla \phi= \int_{B_2r(0)}f \phi^2$$.

Then what is the next step?