Flat solutions of some non-Lipschitz autonomous semilinear equations may   be stable for $N\geq 3$
  We prove that flat ground state solutions ($i.e.$ minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions $N=1,2$ and they can be stable for $N\geq 3$ for suitable values of the involved exponents. 
Jes\'us Ildefonso D\'iaz, Jes\'us Hern\'andez and Yavdat Il'yasov
math.AP