Consider the second order elliptic PDE $$ \begin{cases} Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ on } \partial\Omega\,, \end{cases} $$ where $f\in C^\infty(\Omega)$, $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ is smooth. What are the conditions needed to be put on $L$ and $f$ such that the solution $u$ only has one critical point? I am looking for a result similar to Theorem 2 of this paper(ScienceDirect):

Theorem 2: Let $\Omega$ be a bounded, strictly convex domain in $\mathbb{R}^2$ and $u\in C^3(\Omega)\cap C^1(\bar{\Omega})$ a solution to the boundary value problem \begin{align*} \Delta u &= f(u,\nabla u)\quad\text{ in }\Omega\,,\\ u=\text{const}\,,& \quad\nabla u\neq0\,,\quad\text{ on }\partial\Omega\,, \end{align*} where $f\in C^1$, $f_u\geq0$. Then $u$ has exactly one critical point in $\bar{\Omega}$ and there $\det(D^2u)>0$ holds (i.e., a global maximum or minimum).

I think possibly that if $\nabla f$ is never $0$ that we have a similar statement, but I am not sure.