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

TL;DR

This paper studies positive, non-radially symmetric solutions to exponential elliptic boundary value problems on a solid torus, focusing on existence results in the critical supercritical case and involving Sobolev inequality constants.

Contribution

It establishes the existence of non-radial solutions for exponential elliptic problems on a torus, addressing the supercritical case and symmetry-breaking phenomena.

Findings

Existence of G-invariant solutions without radial symmetry.

Solutions in the supercritical exponential growth regime.

Identification of best Sobolev constants in the critical case.

Abstract

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.