If $u$ is a solution of $\Delta_p u=0$ weakly, then $u^{+}$ is also satisfies $\Delta_p v=0$ weakly. To prove this result, I need to prove that $$ \int_{\Omega}|\nabla v|^{p-2}\nabla v.\nabla \phi\,dx=0 $$ where $v=u^{+}$ for every $\phi\in W_{0}^{1,p}(\Omega)$, whereas I know that $$ \int_{\Omega}|\nabla u|^{p-2}\nabla u.\nabla \phi\,dx=0\:\text{ for every }\:\phi\in W_{0}^{1,p}(\Omega).$$
Can you please help me with this one?