Uniqueness of the critical point for semi-stable solutions in $$\mathbb {R}^2$$

Abstract

In this paper we show the uniqueness of the critical point for semi-stable solutions of the problemwhere ⊂ R 2 is a smooth bounded domain whose boundary has nonnegative curvature and f (0) ≥ 0. It extends a result by Cabré-Chanillo to the case where the curvature of ∂ vanishes.

On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains

“…The result has been recently extended to domains with nonnegative curvature in [7]. We point out that the convexity assumption can not be dropped, indeed for N = 2, for any k ∈ N it is possible to find a smooth "almost convex" domain Ω such that the solution of the torsion problem has at least k critical point, see [9] (see also [6] for a generalization).…”

“…Up to the end of this section let us write u instead of u N for brevity. Let us recall some notations and some results from [4] and [7].…”

“…By the smoothness of u we have that T is of class C 1 . In next lemmas we recall some important properties of the vector field T , proved in [7]. Lemma 2]).…”

On the number of critical points of solutions of semilinear elliptic equations

“…This result was extended to the case of non − negative curvature in [15]. The proof of Theorem 3.3 relies on a careful study of the nodal lines of the derivatives ∂u ∂θ = cos θ ∂u ∂x + sin θ ∂u ∂y , θ ∈ [0, 2π).…”

“…A partial contribution to this question was given in [15] where it was constructed a solution to (1.1) of the torsion problem (3.5) where Ω ⊂ R N , N ≥ 2, its boundary ∂Ω has positive mean curvature and the solution u as k maximum points. Our feeling is that the correct request to extend the Cabré-Chanillo's Theorem is that all the principal curvatures are positive.…”