I'm reading Linear and Nonlinear Functional Analysis by Phillipe Ciarlet, my aim in this book is the Minty-Browder Theorem and their applications. The author shows the following problem
$$\left\{\begin{array}{c} −\Delta_{p} u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.$$
The main idea to solve this problem is use the Minty-Browder Theorem for the operator $\Delta_{p}: W_{0}^{1,p}(\Omega) \rightarrow (W_{0}^{1,p}(\Omega))'$, where $\Delta_{p}(u):= \displaystyle\sum_{i=1}^{N}\dfrac{\partial}{\partial x_{i}}\left(|\nabla u|^{p-2}\dfrac{\partial u}{\partial x_{i}}\right)$. Then, the author says that the duality of p-Laplacian is given by:
$$<\Delta_{p} u, w > = -\displaystyle\int_{\Omega} |\nabla u|^{p-2}u . w $$
My question is: Why? How can I prove thle last equality?
Thanks in advance.
The duality is just integration by parts
$$\langle \Delta_p u,w\rangle = \int_\Omega \text{div} (|\nabla u|^{p-2}\nabla u)\, w \, dx = -\int_\Omega |\nabla u|^{p-2} \nabla u \cdot \nabla w \, dx$$
provided $w\in W_0^{1,p}$. The identity above and Holder's inequality show that
$$|\langle \Delta_p u,w\rangle| \leq C\|\nabla w\|_{L^p(\Omega)} \leq C\|w\|_{W^{1,p}_0},$$
where $C=\||\nabla u|^{p-2}\nabla u\|_{L^q(\Omega)}$ and $\frac{1}{p} + \frac{1}{q} = 1$. Thus $\Delta_p u$ is a bounded linear operator on $W^{1,p}_0$, hence $\Delta_p u \in (W^{1,p}_0(\Omega))'$.