Dear Magda, Beltrami p-Laplace was defined as spherical part of p-Laplace operator on unit sphere to study asymptotic of the solution of corresponding Dirichlet problem in cone. Here is reference  

P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Commun. Partial Differ. Equations 8, 773-817 (1983).

We can take a look, but I'm not sure ifTolksdoff or people before him came from geometrical  formulation. I will ask Sasha Grigoryan if he knows more.

Good, thanks, Akif, I will check the paper.

A few people asked me what I mean by p-Laplace, here it is:

https://en.wikipedia.org/wiki/P-Laplacian

as related to the notion of p-energy.

As for Laplace-Beltrami oper., we all know it:

https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator

I am asking for the perfect hybrid between these 2 notions, on a Riemannian manifold of arbitrary dimension.

What you sent me is a good start and I hope that I can understand the generalization next.

Thanks again!! :-) :-)

Magdalena, the problem is likely of unknown unknowns so that introducing simplicity of an operator runs up to the higher order partial derivatives 3rd, composed within ever expanding spaces, and quantum mechanics might be able to deal with it through relativity, but, honey, we do not know what we do not know: So the Riemannian manifolds keep collapsing and re-emerging as an outcome of time - interference of particles/strings/waves! Truly, you attempt approximating that coffee spill on a blank paper:  So in reality the p-operator of Laplace - Beltrami does not exist... It is merely an  abstract of an illusion of imagination +1 , where matrices of observable dt, are not matrices of observable dt+1

http://rsb5n6.wix.com/projectqrb?fb_ref=Default#!blank/cyhn

And 1 is infinity, !!!! as 0