For bounded domains it follows from the Rellich-Kondrashov compactness theorem $H^1\subset\subset L^p$.

If you denote by $$ S = \inf \frac{\|u\|{H^1}}{\|u\|_{L^p}} $$

and take a sequence $\{u_n\}$ such that $\|u_n\|_{H^1}\to S$ and $\|u_n\|_{L^p} = 1$

then there exists a subsequence $\{u_{n_k}}$ and a function $u$ such that $u_{n_k}\rightharpoonup u$ weakly in $H^1$ and so

$$ \|u\|_{H^1}\le \liminf \|u_{n_k}\|_{H^1} = S $$

Now, you apply R-K Theorem to conclude that $u_{n_k}\to u$ strongly in $L^p$ and so $\|u\|_{L^p}=1$.

By the maximum principle, as you said, it follows that the function $u$ must be positive inside the domain.

For bounded domains it follows from the Rellich-Kondrashov compactness theorem $H^1\subset\subset L^p$.

If you denote by $$ S = \inf \frac{\|u\|{H^1}}{\|u\|_{L^p}} $$

and take a sequence $\{u_n\}$ such that $\|u_n\|_{H^1}\to S$ and $\|u_n\|_{L^p} = 1$

then there exists a subsequence $\{u_{n_k}\}$ and a function $u$ such that $u_{n_k}\rightharpoonup u$ weakly in $H^1$ and so

$$ \|u\|_{H^1}\le \liminf \|u_{n_k}\|_{H^1} = S $$

Now, you apply R-K Theorem to conclude that $u_{n_k}\to u$ strongly in $L^p$ and so $\|u\|_{L^p}=1$.

By the maximum principle, as you said, it follows that the function $u$ must be positive inside the domain.