Estimates of the principal eigenvalue of the \(p\)-Laplacian. (English) Zbl 1245.35075

Summary: We provide estimates from below and from above for the principal eigenvalue of the \(p\)-Laplacian on a bounded domain. We apply these estimates to study the asymptotic behavior of the principal eigenvalue for \(p\to +\infty \).


35P15 Estimates of eigenvalues in context of PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
Full Text: DOI


[1] Drábek, P., Ranges of \(a\)-homogeneous operators and their perturbations, Časopis Pěst. mat., 105, 167-183, (1980) · Zbl 0427.47048
[2] Elbert, Á., A half-linear second order differential equation, (), 153-180
[3] Anane, A., Simplicité et isolation de la première valeur propre du \(p\)-laplacien avec poids, C. R. acad. sci. Paris Sér. I math., 305, 725-728, (1987) · Zbl 0633.35061
[4] Bhattacharya, T., Some results concerning the eigenvalue problem for the \(p\)-Laplacian, Ann. acad. sci. fenn. math., 14, 325-343, (1989) · Zbl 0701.35123
[5] Lindqvist, P., On the equation \(\operatorname{div}(| \nabla u |^{p - 2} \nabla u) + \lambda | u |^{p - 2} u = 0\), Proc. amer. math. soc., 109, 157-164, (1990) · Zbl 0714.35029
[6] 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) · Zbl 1172.35047
[7] Juutinen, P.; Lindqvist, P.; Manfredi, J.J., The \(\infty\)-eigenvalue problem, Arch. ration. mech. anal., 148, 89-105, (1999) · Zbl 0947.35104
[8] Kawohl, B., ()
[9] Allegretto, W.; Huang, Y.X., A picone’s identity for the \(p\)-Laplacian and applications, Nonlinear anal., 32, 819-830, (1998) · Zbl 0930.35053
[10] R.J. Biezuner, J. Brown, G. Ercole, E.M. Martins, Computing the first eigenpair of the \(p\)-Laplacian via inverse iteration of sublinear supersolutions. Preprint arXiv:1011.3172v2. · Zbl 1255.65205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.