It is already known that if $u$ is harmonic in $\Omega\backslash\{x_0\}$ where $\Omega$ is a pre-compact domain in $\mathbb{R}^n$, $n\geq2$ and $u=o(|x-x_0|^{2-n})$ when $x\to x_0$, then the singular point $x_0$ is removable. i.e. $\exists\,\tilde{u}$ harmonic on $\Omega$ such that $\tilde{u}=u$ on $\Omega\backslash\{x_0\}$.
Now I came across a similar but more difficult case. I need to show that the following Dirichlet problem: $$ \Delta u=0\,\,\,\text{in}\,\,\,\Omega\backslash\{x_0\}\\ u|_{\partial\Omega}=f\in C^{\infty}(\mathbb{R})\\ u\in L^{\infty}(\Omega\backslash\{x_0\}) $$
Obviously $f$ does not satisfy the requirements for removable singularities any more. Is it still true that this u coincide with solution of the same Dirichlet problem for Laplace equations on $\Omega$? Thank you in advance.