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Convexity of level curves for solutions to semilinear elliptic equations
Let $\Omega$ be a bounded strictly convex planar domain, and $f$ be a smooth function satisfying $f(0) < 0$ and $f'(t) \geq 0$. In this paper, we provide a simple proof using just the maximum principle that the level curves of the unique positive solution to $\Delta u = f(u)$ in $\Omega$ satisfying $u = 0$ on $\partial\Omega$ are convex and there is a unique critical point. We also provide generalization of this result to cover certain cases with $f'(t) < 0$.