I'm trying to prove the question below. And I'm thinking about using Maximum Principle to prove it. However, U here is not a bounded region. Additionally, for the energy method, I cannot get an idea to apply since integration by parts doesn't work here. Also, there is no condition that the limit of u goes to zero as x goes to infinity...
Could anyone give me some ideas about this question? Thanks a lot!
Let $U = \{x ∈ R^n: |x| > 1\}$, Suppose $u ∈ C^2(U) ∩ C(\overline{U})$ is a bounded solution of the following Dirichlet problem: $∆u = 0$ in $U$ and $u = ϕ$ on $Γ = \{x ∈ R^n: |x| = 1\}$, with $ϕ ∈ C(Γ)$
a) If n = 2, show that there exists at most one solution of the above problem
b) If n=3, show that it is possible to have more than one bounded solutions of the above problem. What additional condition should you impose so that the solution is unique?