Let $B$ denote the unit ball in $ R^N$ centered at the origin and consider $$ -\Delta u(x) + \frac{ x \cdot \nabla u(x)}{|x|^\alpha} = f(x) \quad \mbox{ in } B$$ with $u=0$ on $ \partial B$. (or instead work on an annulus $ \Omega_\epsilon:=\{x \in B: |x|>\epsilon\})$. Here $ \alpha>2$. Here $ f$ is bounded. I seem to be able to get estimates like $ |u(x)| \le C |x|^\alpha$.
QUESTION. Can one expect to get the natural gradient estimate $ | \nabla u(x)| \le C |x|^{\alpha-1}$.
