When is expenditure function non-decreasing?
One possibility is to reparameterize your expenditure as follow: $$e(\mathbf{p},u) = E(f(\mathbf{p}),g(\mathbf{p})u),$$ with $f,g:\mathbb{R}^n\rightarrow \mathbb{R}$. Then use the chain rule to find $\partial{e}/\partial{p_j}$, $\partial{e}/\partial{u}$ and to prove that $e$ is homogeneous of degree one in $\mathbf{p}$ iff ...