Symmetry breaking and other features for eigenvalue problems

In the first part of this paper we discuss a minimization prob- lem where symmetry breaking arise. Consider the principal eigenvalue for the problem −∆u = λχFu in the ball Ba+2 ⊂ RN, where N ≥ 2 and F varies in the annulus Ba+2 \ Ba, keeping a fixed measure. We prove that, if a is large enough, the minimum of the corresponding principal eigenvalue is attained in a subset which is not symmetric. In the second part of the paper we show that the principal eigenvalue of an equation with an indefinite weight in a general bounded domain Ω can be approximated by a functional related to the energy integral of a suitable nonlinear equation; furthermore, we show that a solu- tion of this nonlinear equation approximates, in the H1(Ω) norm, the principal eigenfunction of our problem.