Jiří Benedikt, Department of Mathematics and New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14 Plzeň, Czech Republic, e-mail: benedikt@kma.zcu.cz
Abstract: We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in $\mathbb R^N$ and its asymptotics for $p$ approaching $1$ and $\infty$. Let $p$ tend to $\infty$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty$ for $0<R\leq R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in\mathbb N$ for the $p$-Laplacian and $R_C=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue of the Dirichlet $p$-Laplacian is $NR^{-1}\*(1-(p-1)\log R(p-1))+o(p-1)$ while the asymptotics for the principal eigenvalue of the Navier $p$-biharmonic operator reads $2N/R^2+O(-(p-1)\log(p-1))$.
Keywords: eigenvalue problem for $p$-Laplacian; eigenvalue problem for $p$-biharmonic operator; estimates of principal eigenvalue; asymptotic analysis
Classification (MSC 2010): 35J66, 35J92, 35P15, 35P30
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