Can anyone give an example on the difference between 2-Laplacian and p-Laplacian with p\neq 2? Not only the technical difficulty, but also the essential difference. An example will be much more welcomed if it can be understand in the view of variational method.
Just the number of variables or second order partial derivatives in it, I guess.
The p-Laplacian is the operator that appears in minimizing
integral_D ((1/p)|Grad u|^p -f.u).dx
subject to u = g on the boundary of D. The minimizer is the solution to p-Laplacian(u) = -f with the specified boundary values. The objective function is convex provided p >= 1. The p-Laplacian for p !=2 is a nonlinear 2nd order operator. It is explicitlyl given by the formula
p-Laplacian(u) = div(|grad u|^(p-2).grad u).
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