Energy Stability for a Class of Semilinear Elliptic Problems

In this paper, we consider semilinear elliptic problems in a bounded domain Omega contained in a given unbounded Lipschitz domain C subset of R-N. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain Omega inside C. Once a rigorous variational approach to this question is set, we focus on the cases when C is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.