Let $u$ be a solution to the equation $$\Delta u=f(u,\nabla u)$$ where $f>0$ is a smooth function with $f f_{uu} \le 2f_u^2$. The seminal constant rank theorem states that if $D^2 u$ is positive semidefinite, then $D^2 u$ has constant rank. See the following paper:
[KL] Korevaar, N.J. and Lewis,J.L. Convex solutions of certain elliptic equations have constant rank hessians. Arch.Rational Mech.Anal.(1987) 97: 19. https://doi.org/10.1007/BF00279844.
The result is obtained by pure computation of Laplacian of $\sigma_k(D^2 u)$, and is a consequence of strong maximum principle. My question is, is there any intuitive/geometric explaination why such constant rank theorem holds for solutions of such semilinear equations?
