Exponential elliptic boundary value problems on a solid torus in the critical of supercritical case

In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus T¯ of R3R3, when data are invariant under the group G=O(2)×I⊂O(3)G=O(2)×I⊂O(3). The model problems of interest are stated below:Δυ+γ=f(x)eυ,υ>0 on T,υ|∂T=0 andΔυ+a+feυ=0,υ>0 on T,∂υ∂n+b+geυ=0on ∂T. We prove that exist solutions which are G-invariant and these exhibit no radial symmetries. In order to solve the above problems we need to find the best constants in the Sobolev inequalities in the exceptional case.