What are the basic things that i have to keep in mind to change the PDE having non unique solution to the PDE have unique solution. For example say i have $\Delta u=0 $ for $x\in \Omega$ and $u=1$ in $|x|=1$ Define $\Omega=x: |x| \ge1$
Here we can clearly see that it has infinite solutions . say $u=a(|x|-1)+1$.
How do i argue to put a condition on the given PDE so that the solution turns out unique. Turning it into dirichlet boundary condition gives uniqueness right ? Thanks for help.