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

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- 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 $p\to \infty$.## References

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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