Asymptotics for the principal eigenvalue of the \(p\)-Laplacian on the ball as \(p\) approaches 1. (English) Zbl 1281.35064

Summary: Using the estimates of the principal eigenvalue of the \(p\)-Laplacian on the ball from below and from above, we provide its asymptotic analysis as \(p\to 1_+\).


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
49R05 Variational methods for eigenvalues of operators
33B15 Gamma, beta and polygamma functions
Full Text: DOI


[1] Benedikt, J.; Drábek, P., Estimates of the principal eigenvalue of the \(p\)-Laplacian, J. Math. Anal. Appl., 393, 311-315, (2012) · Zbl 1245.35075
[2] Allegretto, W.; Huang, Y. X., A picone’s identity for the \(p\)-Laplacian and applications, Nonlinear Anal., 32, 819-830, (1998) · Zbl 0930.35053
[3] Drábek, P.; Milota, J., (Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Advanced Texts, (2007), Birkhäuser Basel, Boston, Berlin) · Zbl 1176.35002
[4] Sakaguchi, S., Concavity properties of solutions to some degenerated quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14, 403-421, (1987) · Zbl 0665.35025
[5] 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) · Zbl 1181.35115
[6] Bueno, H.; Ercole, G., Solutions of the Cheeger problem via torsion functions, J. Math. Anal. Appl., 381, 263-279, (2011) · Zbl 1260.49080
[7] Kawohl, B.; Fridman, V., Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin., 44, 659-667, (2003) · Zbl 1105.35029
[8] 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
[9] 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) · Zbl 1255.65205
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