# Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain

I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\mid \nabla u\,\cdot\,\theta = 0\}\,, \quad\theta\in S^1\,,$$ Are smooth curves homeomorphic to $$[0,1]$$ and that the set $$M_\theta = \{x\in N_\theta\mid D^2u\,\cdot\,\theta = 0\}$$ is empty for each $$\theta$$. My question is that wouldn't these statements be true for some smooth function on a convex domain with, say, three nondegenerate critical points? Also why is it necessary for the vector field $$Z$$ to be Lipschitz? To summarise I would like to know what the weakest conditions on $$u$$ are step 2 of the proof.

[1] Cabré, Xavier; Chanillo, Sagun, Stable solutions of semilinear elliptic problems in convex domains, Sel. Math., New Ser. 4, No. 1, 1-10 (1998). ZBL0905.35032.