Symmetry-breaking phenomena in an optimization problem for some nonlinear elliptic equation

Let Omega be a bounded domain in R-n with Lipschitz boundary, lambda>0, and 1less than or equal topless than or equal to(n+2)/(n-2) if ngreater than or equal to3 and 1less than or equal top<+∞ if n=1, 2. Let D be a measurable subset of Ω which belongs to the class C-β={D ⊂ Ω | |D| = β} for the prescribed β ∈ (0, |Ω|). For any D ∈C-β, it is well known that there exists a unique global minimizer u ∈ H-0(1) (Omega), which we denote by u(D), of the functional J(OmegaD)(v) = 1/2 integral(Omega)\delv\(2) dx+lambda/p+1 integral(Omega)\v\(p+1) dx-integral(Omega) chi(D)vdx on H-0(1) (Omega). We consider the optimization problem E-beta,E-Omega = inf(Dis an element ofC beta) J(D)(u(D)) and say that a subset D* is an element ofC(beta) which attains E-beta,E-Omega is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry preserving and symmetry-breaking phenomena of the optimal configuration D* to this optimization problem in various settings.