I have the following energy functional of $p$-Laplacian equation: $$ E(u) = \frac{1}{p} \int_{\Omega} |\nabla u|^p dx $$ for $2.8 \leq p \leq 5$.
My goal is to minimize the energy functional by using nonlinear conjugate gradient method ( https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient_method ) .
To apply nonlinear conjugate gradient method, I have found that
$$ E'(u) = \int_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v dx $$
Now I took five nodes $u_1, u_2, u_3, u_4, u_5$ in one dimension. We know $u_1=0=u_5$ and want to find $u_2,u_3,u_4$. By NCG method we will get $$ \begin{bmatrix} u_2^{n+1}\\ u_3^{n+1}\\ u_4^{n+1} \end{bmatrix} = \begin{bmatrix} u_2^{n}\\ u_3^{n}\\ u_4^{n}\end{bmatrix} - \alpha \cdot \begin{bmatrix} E_2'(u_2^{n})\\ E_3'(u_3^{n})\\ E_4'(u_4^{n}) \end{bmatrix} $$
Is my attempt correct? I am confused about how to manage integral sign of derivative of energy functional ?