I was reading one lecture notes on p-Laplace equation.
Definition: Let $\Omega$ be a domain in $\mathbb{R}^{n}$. We say that $u \in W_{\mathrm{loc}}^{1, p}(\Omega)$ is a weak solution of the $p$ -Laplace equation in $\Omega$, if \begin{equation}\label{eq:1} \int\left\langle|\nabla u|^{p-2} \nabla u, \nabla \eta\right\rangle d x=0 \qquad\qquad (1) \end{equation} for each $\eta \in C_{0}^{\infty}(\Omega) .$ If, in addition, $u$ is continuous, then we say that $u$ is a $p$-harmonic function.
Author given following remark: Remark. If $(1)$ holds for all $\eta \in C_{0}^{\infty}(\Omega)$, then it also holds for all $\eta \in W_{0}^{1, p}(\Omega)$, if we know that $u \in W^{1, p}(\Omega)$.
I do not understand how to prove that. Does this follow from density argument?
Any help or hint will be greatly appreciated.
It is exactly due to density argument. Did you remember how $W_{0}^{1, p}(\Omega)$ is defined? Recall:
Definition: The Sobolev space $W_{0}^{1, p}(\Omega)$ is defined as the completion of $C_{0}^{\infty}(\Omega)$ in the norm of $W^{1, p}(\Omega)$ $$ W_{0}^{1, p}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}^{\|\cdot\|_{W^{1, p_{(\Omega)}}}} $$
What does this mean? It means that if I take a function $f$ in $ W_{0}^{1, p}(\Omega)$, I can always find a sequence of $C_{0}^{\infty}(\Omega)$ functions that converge to $f$ in $\|\cdot\|_{W^{1, p_{(\Omega)}}}$ norm, that means that I can find a $C_{0}^{\infty}(\Omega)$ function "as close as I want" (say $\varepsilon$ > $0$) to $f$ in $\|\cdot\|_{W^{1, p_{(\Omega)}}}$ norm. You see that now, in your definition, if you add and subtract a function $\eta \in W_{0}^{1, p}(\Omega)$, then you can still verify in the limit the equality for all $\eta \in W_{0}^{1, p}(\Omega)$.