I've found many papers characterizing the weak lower semi-continuity of $$ \Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx, $$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$. Are there any results with regards to the same statement with $\Omega$ being all of $\mathbb{R}^d$ itself?

Here $f$ is a non-negative, Borel measurable function, which is convex in its second variable for a.e. one of its first variagbles.