Parameter range below critical exponent

Question

it is known that the critical exponent $\frac{2n}{n+2}$ in the theory of pde often times poses issues, e.g. when considering the $p$-Laplacian. What actually is the meaning of the exponent $p$ in the $p$-Laplacian for $p$ below the critical exponent in the case $n=2$, i.e. $1<p\leq \frac{2n}{n+2} = 1$. Is this simply not possible and only $n\geq 3$ is considered or how is this to understand? Thanks for the help in advance!

Answer 1

Let $n$ be a positive integer and $p\in [1,+\infty)$. The (classical) definition of the $p$-Laplacian is $$\Delta_pu= \operatorname{div} \big ( \vert \nabla u \vert^{p-2} \nabla u \big ) . $$ This definition makes sense for $u \in C^2$ and, additionally, if $p\in [1,2)$ then we require $\vert \nabla u \vert >0$. There is no need for extra constraints on $n$ nor on $p$ (other than the original ones) with respect to the definition.