Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets

Abstract

We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity $e^u$ and the family of shifted power-type nonlinearities $(a+u{)}^p$, where $a>0$ and $p>1$. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.

35 Pages. Dedicated to Xavier Cabré on the occasion of his 60th birthday