On the eigenvalues of the 𝑝-Laplacian with varying 𝑝

Huang, Yin

doi:10.1090/S0002-9939-97-03961-0

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 $p\to \infty$.

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