On continuity of the Gateaux derivative of p-Laplacian operator

Question

Let $\Omega\subset \mathbb{R}^n, N\geq3$, be an open set. For $p\in(1,+\infty)$, define a functional $J:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ by

$J(u)=\int_\Omega |\nabla u|^p\,dx.$

Then $J$ is differentiable in $W_0^{1,p}(\Omega)$ and

$J'(u)v=p\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla v\,dx.$

My question is about how to prove the Gateaux derivative $J_G'(u)$ is continuous. Indeed, for any sequence $u_k\rightarrow u$, I can prove that there exists a subsequence $u_{k_j}$ satisfying $J_G'(u_{k_j})\rightarrow J_G'(u)$ in [$W^{1,p}(\Omega)$]', but I can't prove it for the original sequence.

Thanks for any help!

Answer 1

Use the following convergence principle:

A sequence is converging to some $x$ if and only if every subsequence contains a subsequence that converges to $x$.