radial solutions

Question

I have a question about the following exercise: Find all radial solutions of $$ -\Delta u-x\cdot\nabla u=0,\quad x\in\mathbb{R}^3. $$ How do I express $x\cdot \nabla u$ in spherical coordinates? Thanks!

Answer 1

Let $x=(x_1,\dots,x_n)$. Then $$ \frac{\partial u}{\partial x_i}=\frac{\partial u}{\partial r}\,\frac{\partial r}{\partial x_i}=\frac{x_i}{r}\,\frac{\partial u}{\partial r}. $$ Then $$ x\cdot\nabla u=\sum_{i=1}^nx_i\,\frac{\partial u}{\partial x_i}=\cdots $$