Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.

I am interested in the space of p-harmonic functions on $\Omega$; that is, the metric space $$P = \{ u \in W^{1,p}(\Omega); \Delta_p u = 0 \}.$$ I don't impose any boundary values, so the space is not a singleton.

Has this space been investigated? Is it, for example, a manifold, or does it have some other structure?

If $p=2$, the set $P$ is a linear space, since the Laplacian is linear. Keywords or references even in the linear case would be useful.

This question is inspired by Joonas' question Density of restrictions of $p$-harmonic functions on a hypersurface .