Let $U\subset{\mathbb R}^n$ be a bounded open domain with smooth boundary. I assume that $U$ is diffeomorphic to a ball. You may think of $L=\Delta$ and $U$ is the unit ball.

Let $L={\rm div}(A\nabla\cdot)$ be a uniformly elliptic operator over $U$ with smooth coefficients. I consider solutions of the Dirichlet boundary-value problem
$$Lu=0\quad\hbox{in }\,U,\qquad u=g\quad\hbox{over }\,\partial U.$$

Let me assume that $g$ is a $C^1$-function, with only two critical points $a_\pm$, at which it reaches its minimum and maximum.

From Hopf's Maximum Principle, we know that on the one hand $u$ has no non local minimum/maximum in $U$, and on the other hand $\pm\nabla u(a_\pm)\cdot\vec n>0$, where $\vec n$ is the unit outer normal to $\partial U$. In particular $u$ has no critical point on the boundary.

> Is it true that $u$ has no critical point at all in $U$ ?

Observation : suppose $n=2$, and that $u$ is a Morse function. Then the claim seems to be a consequence of the Morse theory, plus the fact that all critical points have index $1$.