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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the eigenvalues of the $p$-Laplacian with varying $p$
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by Yin Xi Huang PDF
Proc. Amer. Math. Soc. 125 (1997), 3347-3354 Request permission

Abstract:

We study the nonlinear eigenvalue problem \begin{equation*}-\div (| \nabla u|^{p-2} \nabla u)=\lambda |u|^{p-2}u \quad \text {in}\; \Omega , \quad u=0\quad \text {on}\; \partial \Omega ,\tag *{(1) }\end{equation*} where $p\in (1,\infty )$, $\Omega$ is a bounded smooth domain in $\pmb R^{N}$. We prove that the first and the second variational eigenvalues of (1) are continuous functions of $p$. Moreover, we obtain the asymptotic behavior of the first eigenvalue as $p\to 1$ and $p\to \infty$.
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Additional Information
  • Yin Xi Huang
  • Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
  • Email: huangy@mathsci.msci.memphis.edu
  • Received by editor(s): June 14, 1996
  • Additional Notes: Research is partly supported by a University of Memphis Faculty Research Grant
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3347-3354
  • MSC (1991): Primary 35P30, 35B30
  • DOI: https://doi.org/10.1090/S0002-9939-97-03961-0
  • MathSciNet review: 1403133