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

Author: Yin Xi Huang
Journal: Proc. Amer. Math. Soc. 125 (1997), 3347-3354
MSC (1991): Primary 35P30, 35B30
MathSciNet review: 1403133
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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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Yin Xi Huang
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152

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
Article copyright: © Copyright 1997 American Mathematical Society