Stable solutions of semilinear elliptic problems in convex domains

In this note, we consider semilinear equations , with zero Dirichlet boundary condition, for smooth and nonnegative f, in smooth, bounded, strictly convex domains of . We study positive classical solutions that are semi-stable. A solution u is said to be semi-stable if the linearized operator at u is nonnegative definite. We show that in dimension two, any positive semi-stable solution has a unique, nondegenerate, critical point. This point is necessarily the maximum of u. As a consequence, all level curves of u are simple, smooth and closed. Moreover, the nondegeneracy of the critical point implies that the level curves are strictly convex in a neighborhood of the maximum of u. Some extensions of this result to higher dimensions are also discussed.