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Remarks on the critical points of solutions to some quasilinear elliptic equations of second order in the plane. (English) Zbl 0808.35029
This paper adds new information concerning critical points of a solution $$u\in C^ 3 (\Omega)\cap C^ 1 (\overline{\Omega})$$ of a semilinear elliptic equation $$\Delta u= f(u, \nabla u)$$ in a two-dimensional bounded domain $$\Omega$$, where $$f\in C^ 1$$ and $$f_ u\geq 0$$. Under the assumption that the mapping $$\nabla u/ |\nabla u|$$ is injective on $$\partial\Omega$$, the number of critical points in $$\overline{\Omega}$$ is finite and odd, and each critical point is nondegenerate. Especially, in a strictly convex domain $$\Omega$$, $$u$$ has exactly one critical point in $$\overline {\Omega}$$ provided that $$u$$ is constant and $$\nabla u$$ never vanishes on $$\partial\Omega$$. The proof is based on a careful analysis of the nodal lines of directional derivatives of $$u$$, which seems peculiar to the two-dimensional case.

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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