A question related to Laplace equation on pde.

Question

Let $B_R(p)$ the open ball of radius R centered at p in $R^n$ and consider the following problem $∆u = 0 $ in $ R^n\setminus B¯_R(p)$ ,$u = c$ on $∂B_R(p)$, where c is a given constant. Find a non-constant solution $u\in C^2(R^n\setminus B¯R(p))$ of the above problem. I think we get to use Laplace equation.I am not able to figure out how to get such non-constant solution.

Answer 1

Look for a solution with radial simmetry, that is $u(x)=u(r)$, where $r=|x|=\sqrt{x_1^2+\dots+x_n^2}$. The equation becomes $$ \frac{d^2u}{dr^2}+\frac{n-1}{r}\,\frac{du}{dr}=0,\quad r>R,\quad u(R)=c. $$