# Article

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Keywords:
eigenvalue problem for $p$-Laplacian; eigenvalue problem for $p$-biharmonic operator; estimates of principal eigenvalue; asymptotic analysis
Summary:
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 $2N/R^2+O(-(p-1)\log (p-1))$.
References:
[1] Allegretto, W., Huang, Y. X.: A Picone's identity for the $p$-Laplacian and applications. Nonlinear Anal., Theory Methods Appl. 32 819-830 (1998). MR 1618334 | Zbl 0930.35053
[2] Benedikt, J., Drábek, P.: Asymptotics for the principal eigenvalue of the $p$-biharmonic operator on the ball as $p$ approaches 1. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 95 735-742 (2014). DOI 10.1016/j.na.2013.10.016 | MR 3130558 | Zbl 1281.35061
[3] Benedikt, J., Drábek, P.: Asymptotics for the principal eigenvalue of the $p$-Laplacian on the ball as $p$ approaches 1. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 93 23-29 (2013). DOI 10.1016/j.na.2013.07.026 | MR 3117145 | Zbl 1281.35064
[4] Benedikt, J., Drábek, P.: Estimates of the principal eigenvalue of the $p$-biharmonic operator. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 5374-5379 (2012). DOI 10.1016/j.na.2012.04.055 | MR 2927595 | Zbl 1244.35096
[5] Benedikt, J., Drábek, P.: Estimates of the principal eigenvalue of the $p$-Laplacian. J. Math. Anal. Appl. 393 311-315 (2012). DOI 10.1016/j.jmaa.2012.03.054 | MR 2921671 | Zbl 1245.35075
[6] Biezuner, R. J., Brown, J., Ercole, G., Martins, E. M.: Computing the first eigenpair of the $p$-Laplacian via inverse iteration of sublinear supersolutions. J. Sci. Comput. 52 180-201 (2012). DOI 10.1007/s10915-011-9540-0 | MR 2923523 | Zbl 1255.65205
[7] Biezuner, R. J., Ercole, G., Martins, E. M.: Computing the first eigenvalue of the $p$-Laplacian via the inverse power method. J. Funct. Anal. 257 243-270 (2009). DOI 10.1016/j.jfa.2009.01.023 | MR 2523341 | Zbl 1172.35047
[8] Bueno, H., Ercole, G., Zumpano, A.: Positive solutions for the $p$-Laplacian and bounds for its first eigenvalue. Adv. Nonlinear Stud. 9 313-338 (2009). DOI 10.1515/ans-2009-0206 | MR 2503832 | Zbl 1181.35115
[9] Drábek, P., Milota, J.: Methods of Nonlinear Analysis. Applications to Differential Equations. Birkhäuser Advanced Texts: Basel Lehrbücher Birkhäuser, Basel (2007). MR 2323436 | Zbl 1176.35002
[10] Drábek, P., Ôtani, M.: Global bifurcation result for the $p$-biharmonic operator. Electron. J. Differ. Equ. (electronic only) 2001 48 19 pages (2001). MR 1846664 | Zbl 0983.35099
[11] Jaroš, J.: Picone's identity for the $p$-biharmonic operator with applications. Electron. J. Differ. Equ. (electronic only) 2011 122 6 pages (2011). MR 2836803 | Zbl 1229.35024
[12] Juutinen, P., Lindqvist, P., Manfredi, J. J.: The $\infty$-eigenvalue problem. Arch. Ration. Mech. Anal. 148 89-105 (1999). DOI 10.1007/s002050050157 | MR 1716563 | Zbl 0947.35104
[13] Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics 1150 Springer, Berlin (1985). DOI 10.1007/BFb0075060 | MR 0810619 | Zbl 0593.35002
[14] Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant. Commentat. Math. Univ. Carol. 44 659-667 (2003). MR 2062882 | Zbl 1105.35029

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