zbMATH — the first resource for mathematics

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.

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