$H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$?

Question

In a paper I see that the authors used $H_{0}^{1}(\Omega)\hookrightarrow L^4(\Omega)$ where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary. I think that this imbedding holds only for those $N$ which satisfy $4\leq 2^\star$($2^\star=\frac{2N}{N-2}$) i.e. when $\Omega$ is in $\mathbb{R}^3$ or $\mathbb{R}^4$. How we can show this imbedding for higher dimensions?

Answer 1

The general result is called the Sobolev embedding theorem, which states that $H_0^k$ is embedded in $L^q$ where $$\frac{1}{q}=\frac{1}{2}-\frac{k}{n}$$ where $n$ is the dimension of the space. This can be generalized even further, see the Wikipedia article.