Let $B\colon W^{1,p}\times W^{1,q}\rightarrow \mathbb{R}$ and $\frac{1}{p}+\frac{1}{q}=1$ , $p>2$ where
$$B[u,v]=\int \left( Du \,Dv+ h u v \right)\, dx$$
where $h>0$. I am interested to prove
$$\sup_{\substack{v\in W^{1,q}\\ v \neq 0}}\frac{B[u,v]}{\|v\|_{W^{1,q}}}\geq C \|u\|_{W^{1,p}}$$
Would someone please give me some hint!
Thanks.