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On the eigenvalues of the $p$-Laplacian with varying $p$

Author(s): Yin Xi Huang
Journal: Proc. Amer. Math. Soc. 125 (1997), 3347-3354.
MSC (1991): Primary 35P30, 35B30
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Abstract: We study the nonlinear eigenvalue problem

\begin{equation*}-\text {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

DOI: 10.1090/S0002-9939-97-03961-0
PII: S 0002-9939(97)03961-0
Keywords: Eigenvalues, the $p$-Laplacian
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 of article: Copyright 1997, American Mathematical Society

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