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 nontrivial positive solution to the nonlinear Neumann problem $$displaylines{ Delta_qu+a(x)u^{q-1}=lambda f(x)u^{p-1}, quad u>0quad hbox{on } T,cr abla u|^{q-2}frac{partial u}{partial u}+b(x) u^{q-1} =lambda g(x)u^{ilde{p}-1} quadhbox{on }{partial T},cr p =frac{2q}{2-q}>6,quad ilde{p}=frac{q}{2-q}>4,quad frac{3}{2}<q<2, }$$ on a solid torus of $mathbb{R}^3$. When data are invariant under the group $G=O(2)imes I subset O(3)$, 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).

Keywords

• Neumann problem
• q-Laplacian
• solid torus
• no radial symmetry
• critical of supercritical exponent