Practical applications of the $L^p$ norm when $p \neq 1,2,\infty$
One example not mentioned in the MO thread: the $L^4$ norm appears in the Ginzburg-Landau formula for the energy of superconductor.
More generally: we want nonlinear variational models to describe phenomena which do not obey linear superposition. Minimization of $L^2$ norm leads to linear PDE. Minimization of $L^1$ and $L^\infty $ norms leads to badly degenerate things that are barely PDE at all. The norms $L^4$ and $L^{\rm dimension}$ emerge as attractive alternatives.