Symmetry breaking in problems involving semilinear equations

CADEDDU, LUCIO;Porru G.

2011-01-01

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Abstract

This paper is concerned with two maximization problems where symmetry breaking arises. The first one consists in the maximization of the energy integral relative to a homogeneous Dirichlet problem governed by the elliptic equation −∆u = χF uq in the annulus Ba,a+2 of the plane. Here 0 ≤ q < 1 and F is a varying subset of Ba,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 Ba+2 when F varies in the annulus Ba,a+2 keeping a fixed 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.