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 $H^{\bullet}(V, j_{!*}L)$?
In general the "attachment triangle" gives you a long exact sequence $$ \ldots \to H^{k}(Z,i^! j_{!\ast} L) \to H^k(V,j_{!\ast} L) \to H^k(U,L) \to H^{k+1}(Z,i^! j_{!\ast} L) \to \ldots $$ where $i \colon Z \to V$ is the inclusion of the complement of $U$. Using this you can derive various conditions implying the nonvanishing of $H^k(V,j_{!\ast} L)$.
For instance, suppose that $L$ is mixed of weights $\geq w$. Then the same is true for $i^! j_{!\ast} L$. Thus if $W_{k+w} H^k(U,L) \neq 0$ then $H^k(V,j_{!\ast} L)$ must also be nonzero, since $H^{k+1}(Z,i^! j_{!\ast} L)$ is mixed with weights $\geq k+1+w$.
