# existence of a harmonic function

Let $\Omega\subset\mathbb R^n$ open, not bounded and $n\ge3$. Let $\partial\Omega$ bounded and regular concering the laplace operator. Given a continuous function $\phi:\partial\Omega\rightarrow\mathbb R$ and $\gamma\in\mathbb R$ there exists a harmonic function $u\in C^2(\Omega)\cap C^0(\overline\Omega)$ with $u=g\space\space\text{on}\space\partial\Omega$ and $\lim\limits_{|x|\rightarrow\infty}u(x)=\gamma$

How can you prove the existence?

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You question is a little confusing to me because you say that $\Omega$ is not bounded, but $\partial\Omega$ is. Both references I gave only deal with the case where the region is bounded. However, I have no doubt that armed with the magic search term "Dirichlet Problem," you will be able to find literature about the unbounded case. This is a classic problem, and I'm sure others can give more authoritative references if you are looking for something research-level.
Complex analysis is not very helpful here because the existence result stated by @Niro is false for $n=2$. –  user31373 Jun 15 '12 at 1:34