A Neumann problem with the q-Laplacian on a solid torus in the critical of supercritical case
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Following the work of Ding  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).
CitationCotsiolis, A., & Labropoulos, N. (2007). A Neumann problem with the q-Laplacian on a solid torus in the critical of supercritical case. Electronic Journal of Differential Equations, 2007(164), pp. 1-18.
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