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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.

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