A Neumann problem with the q-Laplacian on a solid torus in the critical of supercritical case

Abstract

Following the work of Ding [21] we study the existence of a non-trivial positive solution to the nonlinear Neumann problem Δqu + α(x)uq-1 = λƒ(x)up-1, u > 0 on T, ∇u|q-2 ∂u/∂v + b(x)uq-1 = λg(x)up̃-1 on ∂T, p = 2q/2-q > 6, p̃ = q/2-q > 4, 3/2 < q < 2, on a solid torus of ℝ3. When data are invariant under the group G = O(2) x I ⊂ O(3), we find solutions that exhibit no radial symmetries. First we find the best constants in the Sobolev inequalities for the supercritical case (the critical of supercritical).