given the equation
$$ 0 = \int_\Omega \Delta (p ^m) v dx $$
with $\Omega \subset \mathbb{R}^2$ and assuming zero boundary conditions.
Is it possible to write something like
$$ 0 = \int_\Omega \nabla (p ^m) \nabla v dx $$
using the divergence theorem?
This is true if you assume that either $v$ or $\nabla p^m$ vanish on the boundary. You have $$ \nabla\cdot(v\nabla p^m)=v\Delta p^m+\nabla v\cdot\nabla p^m $$ Integrating over $\Omega$, the left hand side can be transformed using Stokes' Thm $$ \int\limits_{\Omega}\nabla\cdot(v\nabla p^m)dx=\int\limits_{\partial\Omega}v\nabla p^m\cdot dS $$ which vanishes if either $v$ or $\nabla p^m$ vanish on the boundary.Therefore, you are left with $$ \int\limits_{\Omega}v\Delta p^m\,dx=-\int\limits_{\Omega}\nabla v\cdot\nabla p^m\,dx $$ and if one of them vanishes, the other one vanishes as well.