Remarks on the critical points of solutions to some quasilinear elliptic equations of second order in the plane. (English) Zbl 0808.35029

Pagani-Masciadri, K. F.

J. Math. Anal. Appl. 174, No. 2, 518-527 (1993).

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.

Reviewer: K.-I.Nagasaki (Minneapolis)

MSC:

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

Keywords:

critical points of a solution; semilinear elliptic equation; two- dimensional bounded domain