I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$ E[u]=\frac{1}{p}\int_\Omega \lvert\nabla u\lvert^pdx+\int_\Omega fu\,dx $$ where $\Omega$ is open bounded nice boundary and $f\in W^{-1,p'}(\Omega)$. It is clear that $E[u]$ has a unique minimizer $\bar u$ by using the direct method. So far so good.
But next, my textbook states that for $p\geq 2$ the $p$-Laplacian is strongly monotone in the sense that $$ \int_\Omega (\lvert \nabla \bar u\lvert^{p-2}\nabla \bar u-\lvert \nabla v\lvert^{p-2}\nabla v)(\nabla \bar u-\nabla v)dx\geq C\|\bar u-v\|_{W_{0}^{1,p}(\Omega)}^p \tag 1$$ for all $v\in W_0^{1,p}(\Omega)$
The book does not give any prove of $(1)$. It looks to me that this is somehow an variational inequality but I can not prove it either... Please help me.
Also, the book also states that by strongly monotone then the solution is unique. I can not see why monotone implies uniqueness of solution... (The way I see the solution is unique is that $E[u]$ is strictly convex..)
Any help is really welcome! Happy new year guys, by the way. :)