I want to ask a question about the standard approximation argument while reading something about stable solution of elliptic PDE Remark 1.1.1.
We set the following definition, we say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it satisfies the following definition. Here $u \in C^{\infty}(\Omega)$ and $f \in C^1(\Omega)$. $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall \varphi \in C_c^1(\Omega), $$ then using a standard approximation argument, if $\Omega$ is bounded or merely if $f^{\prime}(u)^{-}$ is bounded in $\Omega$, one can take $\varphi \in H_0^1(\Omega)$ in the above definition.
I want to ask
1.If $\Omega$ is bounded then function in $W_0^{1,2}(\Omega)$ can be approximated by a sequence of functions in $C_c^{\infty}(\Omega)$, besides, since $\Omega$ is bounded, so $f^{\prime}(u)$ is also bounded, then the approximation is simple.
2.If $\Omega$ is unbounded, then function in $W_0^{1,2}(\Omega)$ can also be approximated by a sequence of functions in $C_c^{\infty}(\Omega)$ ? I can't find appropriate reference, so I'm not sure if this is right.
Besides, for $\Omega$ unbounded, why we only want $f^{\prime}(u)^{-}$ to be bounded? $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} (f^{\prime}(u)^+-f^{\prime}(u)^{-}) \varphi^2 d x \geq 0, \quad \forall \varphi \in C_c^1(\Omega), $$ if we want the approximation, don't we need that $|f^{\prime}(u)|$ is bounded?