I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, or $\Omega$ is the light cone.

My question is the existence of solution to this problem.

I understand that this problem has been studied extensively, there should be many results. Yet I still fail to find the answer. So I hope someone can give me a summary of this problem. Thanks!