I have the following problem:
$\Delta u =0$ in $\Omega_e = \mathbb{R}^3 - \overline{\Omega}$, and with condiction $u=0$ on $\partial \Omega$ and $u=o(1)$, that is $\lim_{r \rightarrow 0} u(x) =0$. Where $u=u(|x|)=u(r).$
Is it correct to say that this problem has no unique solution?
For example, for $\Omega=B(0,1)$, I can consider $u(x)=1 - 1/|x|$ as solution, but also $u(x)=0$ is a solution. Is this correct?
Uniqueness holds, due to the condition that $u=o(1)$ at infinity (that is, $u(x)\to 0$ as $|x|\to\infty$). Your function $1-1/|x|$ does not satisfy that.
To show uniqueness, let $u$ be the difference of two solutions. Given $\epsilon>0$, there is $R$ such that $|u|<\epsilon$ on the sphere $|x|=R$. By the maximum principle applied in the domain $\{x : |x|<R\} \cap \Omega_e$, we have $|u|<\epsilon$ in $G_R$. Since $\epsilon $ was arbitrary, the conclusion follows.