Let $V$ be a $k$-vector space and $U$ an open subscheme of $V$.

  Let $L$ be an $l$-adic local system on $U$, under which conditions the non-vanishing of the cohomology of $H^{\bullet}(U,L)$ implies the non-vanishing of the cohomology $H^{\bullet}(V, j_{!*}L)$ where $j:U\rightarrow V$ is an open immersion.

Let $V$ be affine $n$-space over a field $k$; and $j \colon U \to V$ an open subscheme of $V$. Let $L$ be an $\ell$-adic local system on $U$.

Suppose the cohomology of $H^{\bullet}(U,L)$ does not vanish.

Are there conditions that imply the non-vanishing of the cohomology $H^{\bullet}(V, j_{!*}L)$?