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Proceedings of the American Mathematical Society

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Asymptotics of Sobolev embeddings and singular perturbations for the $p$-Laplacian

Authors: Manuel del Pino and César Flores
Journal: Proc. Amer. Math. Soc. 130 (2002), 2931-2939
MSC (2000): Primary 35J20; Secondary 35B40
Published electronically: April 10, 2002
MathSciNet review: 1908916
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Abstract: We consider the best constant $S(\Omega_\lambda)$for the embedding of $W^{1,p} (\Omega_\lambda)$ into $L^q(\Omega_\lambda)$ where $1<p<2$, $p<q< {Np\over N-p}$. Here $\Omega_\lambda = \lambda \Omega$ with $\Omega$a smooth, bounded domain in $\mathbb{R} ^n$ and $\lambda$ a large positive number. It is proven by the validity of the expansion

\begin{displaymath}S( \Omega_\lambda) = S(\mathbb{R} ^n_+) - \lambda^{-1} \gamma \max_{x\in \partial \Omega} H(x) + o ( \lambda^{-1} ), \nonumber\end{displaymath}  

as $\lambda \to \infty$, where $\gamma$ is a positive constant depending on $p,q$ and $N$. The behavior of associated extremals, which satisfy an equation involving the $p$-Laplacian operator, is also analyzed.

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Additional Information

Manuel del Pino
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMR2071 CNRS-UChile), Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

César Flores
Affiliation: Departamento de Matemáticas, FCFM Universidad de Concepción, Casilla 160-C, Concepción, Chile

Received by editor(s): May 1, 2001
Published electronically: April 10, 2002
Additional Notes: This work was supported by grants Fondecyt Lineas Complementarias 8000010, DIUC 200.015.015-1.0, ECOS/CONICYT, and FONDAP
Dedicated: To the memory of Carlos Cid
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society