Electron. J. Diff. Eqns., Vol. 2007(2007), No. 164, pp. 1-18.

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

Athanase Cotsiolis, Nikos Labropoulos

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 R3. 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.

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Athanase Cotsiolis
Department of Mathematics, University of Patras
Patras 26110, Greece
email: cotsioli@math.upatras.gr
Nikos Labropoulos
Department of Mathematics, University of Patras
Patras 26110, Greece
email: nal@upatras.gr

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