TITLE

# Poincarï¿½ inequality and Palaisï¿½Smale condition for the p-Laplacian

AUTHOR(S)
Dr�bek, Pavel; Tak�c, Peter
PUB. DATE
May 2007
SOURCE
Calculus of Variations & Partial Differential Equations;May2007, Vol. 29 Issue 1, p31
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
An improved Poincarï¿½ inequality and validity of the Palais-Smale condition are investigated for the energy functional $$\mathcal{J}_{\lambda}(u) = \frac{1}{p} \int\limits_{\Omega}|\nabla u|^{p} {\rm d}x - \frac{\lambda}{p} \int\limits_{\Omega} |u|^{p} {\rm d}x - \int\limits_{\Omega} fu {\rm d}x$$ on $$W_{0}^{1,p}(\Omega)$$ , 1 < p < 8, where O is a bounded domain in $$\mathbb{R}^{N}$$ , $$\lambda\in \mathbb{R}$$ is a spectral (control) parameter, and $$f \in L^{\infty}(\Omega)$$ is a given function, $$f \not\equiv 0$$ in O. Analysis is focused on the case ? = ?1, where -?1 is the first eigenvalue of the Dirichlet p-Laplacian ? p on $$W_{0}^{1,p}(\Omega)$$ , ?1 > 0, and on the ï¿½quadratizationï¿½ of $$\mathcal{J}_{\lambda_1}$$ within an arbitrarily small cone in $$W_{0}^{1,p}(\Omega)$$ around the axis spanned by $$\varphi_1$$ , where $$\varphi_1$$ stands for the first eigenfunction of ? p associated with -?1.
ACCESSION #
24152091

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