Symmetry properties for the extremals of the Sobolev trace embedding - Bonder, J.F. (2004)
In this article we study symmetry properties of the extremals for the Sobolev trace embedding H1(B(0,μ))→Lq(∂B(0, μ)) with 1≤q≤2(N-1)/(N-2) for different values of μ. These extremals u are solutions of the problem Δu=uinB(0,μ),∂u∂η= λ|u|q-2uon∂B(0,μ). We find that, for 1≤q<2(N-1)/(N-2), there exists a unique normalized extremal u, which is positive and has to be radial, for μ small enough. For the critical case, q=2(N-1)/(N-2), as a consequence of the symmetry properties for small balls, we conclude the existence of radial extremals. Finally, for 1<q≤2, we show that a radial extremal exists for every ball. © 2004 Elsevier SAS. All rights reserved.