Symmetry breaking in problems involving semilinear equations

This paper is concerned with two maximization problems where symmetry breaking arises. The rst one consists in the maximization of the energy integral relative to a homogeneous Dirichlet problem governed by the elliptic equation -$\deltau = XF^u^q$ in the annulus $B_(a,a+2)$ of the plane. Here 0  q < 1 and F is a varying subset of $B_(a,a+2)$, with a fixed measure. We prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough. The second problem we consider is governed by the same equation in a disc $B_a+2$ when $F$ varies in the annulus Ba;a+2 keeping a xed measure. So, now we have a so called maximization problem with a constraint. As in the previous case, we prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough.