We study the existence of symmetric ground states to the supercritical
problem −Δv=λv+∣v∣p−2v in Ω,v=0 on ∂Ω, in a domain of the form Ω={(y,z)∈Rk+1×RN−k−1:(∣y∣,z)∈Θ}, where Θ is a bounded smooth
domain such that Θ⊂(0,∞)×RN−k−1,1≤k≤N−3,λ∈R, and
p=N−k−22(N−k) is the (k+1)-st critical exponent. We show that
symmetric ground states exist for λ in some interval to the left of
each symmetric eigenvalue, and that no symmetric ground states exist in some
interval (−∞,λ∗) with λ∗>0 if k≥2.
Related to this question is the existence of ground states to the anisotropic
critical problem −div(a(x)∇u)=λb(x)u+c(x)∣u∣2∗−2uinΘ,u=0on∂Θ, where a,b,c are positive continuous functions on
Θ. We give a minimax characterization for the ground states
of this problem, study the ground state energy level as a function of
λ, and obtain a bifurcation result for ground states