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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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