article

Title: Best constants in some exponential Sobolev inequalities

Authors: Bernd Kawohl and Marcello Lucia

Issue: Volume 57 (2008), Issue 4, 1907-1928

Abstract:

$A Pohozaev identity is used to classify the radial solutions of a quasilinear equation with exponential nonlinearity. The results are applied to find the infimum of the non-local functional \[ \mathcal{F}(\lambda,u)=\frac{1}{n}\int_{\Omega}|\nabla u|^n\,\mathrm{d}x-\lambda F\bigg(\barint{\Omega}e^u\,\mathrm{d}x\bigg),\quad u\in W^{1,n}_0(\Omega), \] for various nonlinearities $F$, where $\Omega$ is a bounded domain of $\mathbb{R}^n$ and $\lambda$ a real parameter. Our results generalize the case when $F(s)=\log s$.$