Could you tell me the physics background of pLaplacian equation? Thank you! Actually, I know nothing about this. But I am curious about the original of these PDEs or where they come from. Could you introduce some related books or articles to me?

It is used to model Newtonian and non Newtonian Fluids. You generally have different values for $p$ depending on the type of fluid:
Edit: I just realized that this alone probably doesn't satisfy your question. Let me clearify the actual physics behind the above thinking: The viscous property of those fluids arises from shear stress ("drag") inside of the fluid. To describe this effect in fluids, we use the speed the particles travel in relation to each other (the factor being $\vert \nabla u \vert ^{p2}$ ). Newtonian fluids are purely linear, so the correct approximation is $p=2$. For nonnewtonian the viscosity is nonlinear, so $p\not= 2$. 

