Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 164, pp. 1-18.

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

Authors: Athanase Cotsiolis (Univ. of Patras, Greece)
         Nikos Labropoulos (Univ. of Patras, Greece)

Abstract: 
 Following the work of Ding [21] we study the existence of
 a nontrivial positive solution to the nonlinear Neumann problem
$$\displaylines{
 \Delta_qu+a(x)u^{q-1}=\lambda f(x)u^{p-1}, \quad u>0\quad \hbox{on } T,\cr
 \nabla u|^{q-2}\frac{\partial u}{\partial \nu}+b(x) u^{q-1}
 =\lambda g(x)u^{\tilde{p}-1} \quad\hbox{on }{\partial T},\cr
 p =\frac{2q}{2-q}>6,\quad
 \tilde{p}=\frac{q}{2-q}>4,\quad  \frac{3}{2}<q<2,
}$$
 on a solid torus of $\mathbb{R}^3$.  When data are invariant under
 the group  $G=O(2)\times I \subset O(3)$, 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).
  
Submitted May 18, 2006. Published November 30, 2007.
Math Subject Classifications: 35J65, 46E35, 58D19.
Key Words: Neumann problem; q-Laplacian; solid torus; no radial symmetry;
          critical of supercritical exponent.