I am reading chapter 14 of Michael E. Taylor's book, Partial Differential Equations. I am confused because I can't seem to find what it means for a fully nonlinear PDE to be elliptic.
Here is the setup to my question: suppose $f=f(x,\zeta)$ is smooth in its arguments, for $x\in \mathbb{R}^n$, $\zeta = \{\zeta_\alpha \colon |\alpha|\leq m \}$ and $\alpha$ is a multiindex. The pde to be solved can be written as $$ f(x,D^m u)=g(x), \quad (*) $$ where $D^mu =\{ D^\alpha u \colon |\alpha|\leq m\}$ stands for all combinations of partial derivatives of order less than $m$. Now, we can use this to build a non-linear differential operator $F: C^\infty \to C^\infty$ given by $F(u) = f(-, D^mu)$.
Taylor writes that the operator $F$ is elliptic at $u_0$ provided its linearization $DF_{u_0}$ is a linear, elliptic differential operator. This is all well and good.
However, later in this chapter, Taylor refers to the PDE $(*)$ as being elliptic. In particular, I would like to use regularity results such as Theorem 4.6 in this same chapter, and Taylor refers to having a (weak?) solution to an elliptic PDE. So, my question really is
What exactly is meant here by the equation $(*)$ being elliptic?
It seems like there can be multiple meanings. If I want the regularity result, do I need the associated non-linear $F$ to be elliptic at the relevant weak solution? Do I need ellipticity of $F$ in a neighborhood of a weak solution? My apologies if my poor reading skills are to blame for my understanding here.