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

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

2009-12-18

arxiv.org/abs/0912.3558v2

J. Differential Equations 251 (2011) 651-687

Mathematics

Analysis of PDEs

Scientific paper

10.1016/j.jde.2011.04.019

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 $\bar{T}$ of $\mathbb{R}^3$, when data are invariant under the group $G=O(2)\times I \subset O(3)$. The model problems of interest are stated below: ${ll} {\bf(P_1)} & \displaystyle \Delta\upsilon+\gamma=f(x)e^\upsilon, \upsilon>0\quad \mathrm{on} \quad T, \quad\upsilon |_{_{\partial T}}=0.$ and ${ll}\bf{(P_2)} & \displaystyle \Delta\upsilon+a+fe^\upsilon=0, \upsilon>0\quad \mathrm{on}\quad T, [1.3ex] &\displaystyle \frac{\partial \upsilon}{\partial n}+b+ge^\upsilon=0\quad \mathrm{on} \quad{\partial T}.$ 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.

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