Let $W^{1, p}(\Omega; \mathbb{R})$ be the standard Sobolev space and $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. If $f\colon\Omega\times\mathbb{R}\times\mathbb{R}^2$ is a Caratheodory integrand, it is known that convexity of $f(\mathbf{x},u,\cdot)$ for every $(\mathbf{x}, u)\in\Omega\times\mathbb{R}$ is a sufficient condition for weak lower semicontinuity for functionals of the form $$ F(u) = \int_\Omega f(\mathbf{x}, u(\mathbf{x}), \nabla u(\mathbf{x}))\, d\mathbf{x}, \ \ u(\mathbf{x}) = u(r, z)\in W^{1, p}(\Omega; \mathbb{R}). $$
Does this result generalize to weighted Sobolev spaces $W_r^{1, p}(\Omega; \mathbb{R})$? $$ W_r^{1, p}(\Omega;\mathbb{R}) = \left\{u\colon \int_\Omega \left(|u|^p + |\nabla u|^p\right)r\, d\mathbf{x} < \infty\right\} $$