I was reading some research articles concerning the Liouville's type theorem in PDEs (Almost all of them are entitled Liouville's type theorem for ...). Here is the statement of one of those theorem for the stationary magneto-hydrodynamics system (MHD)
Let $(u, H)$ be a smooth solution of system (MHD) with $u,~ H \in BMO^{−1}(\mathbb{R}^3)$. If we additionally require that $u,~ H \in L^q(\mathbb{R}^3)$ for $2<q<6$, then $u \equiv 0$ and $H \equiv 0$.
Can we confirm that this type of theorems (The Liouville's type theorems) are named after the well known Liouville theorem in complex analysis which reads
Every holomorphic bounded function $f$, i.e.: there exists a positive number $M$ such that ${\displaystyle |f(z)|\leq M}$ for all ${\displaystyle z} \in {\displaystyle \mathbb {C} }$ is constant.
In other words, in PDEs theory we can call a Liouville's type theorem every result that states:
If a function $f$ belongs to a specific functional (class/spaces), then $f$ is either constant or identically zero.
It is close. I would say that a Liouville-type theorem is one that says that all solutions of a given PDE in a certain function space are constant. For many function spaces, zero is of course the only constant function that belongs to them.
In the classical Liouville theorem you can take as PDE the Laplace equation for the real and imaginary part and as function space $L^\infty$. I would go so far to say that the classical model for these Liouville-type theorems is the result that states that every bounded harmonic function in $\mathbb{R}^n$ is constant. That these solutions happen to be real and imaginary parts of holomorphic functions on $\mathbb{C}$ is just a nice coincidence and a theorem that happens to have a name.