HARDY-SOBOLEV-MAZ'YA INEQUALITIES: SYMMETRY AND BREAKING SYMMETRY OF EXTREMAL FUNCTIONS

Denote points in R^k ×R^{N - k} as pairs ξ = (x,y), and assume 2 ≤ k < N. In this paper, we study the problem -Δ v=λ|x|^{-2} v+ |x|{-b}v^{p-1} in R^N, x≠ 0, ν > 0 where $p > 2, b = N - pN - 2\2 and λ ≤ (k-2\2)2, the Hardy constant. We prove existence, symmetry and breaking symmetry results.