×

Sublinear eigenvalue problems associated to the Laplace operator revisited. (English) Zbl 1221.35270

Summary: Eigenvalue problems involving the Laplace operator on bounded domains lead to a discrete or a continuous set of eigenvalues. In this paper, we highlight the case of an eigenvalue problem involving the Laplace operator which possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, at least one more eigenvalue which is isolated in the set of eigenvalues of that problem.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, Comptes Rendus Mathématique. Académie des Sciences. Paris, Sér. I Math. 305 (1987), 725–728. · Zbl 0633.35061
[2] H. Brezis, Analyse Fonctionelle. Théorie et Applications, Collection Mathématiques Appliqu ées pour la Maîtrise, Masson, Paris, 1983.
[3] X. Fan, Remarks on eigenvalue problems involving the p(x)-Laplacian, Journal of Mathematical Analysis and Applications 352 (2009), 85–98. · Zbl 1163.35026 · doi:10.1016/j.jmaa.2008.05.086
[4] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, Journal of Mathematical Analysis and Applications 302 (2005), 306–317. · Zbl 1072.35138 · doi:10.1016/j.jmaa.2003.11.020
[5] R. Filippucci, P. Pucci and V. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Communications in Partial Differential Equations 33 (2008), 706–717. · Zbl 1147.35038 · doi:10.1080/03605300701518208
[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998. · Zbl 1042.35002
[7] P. Lindqvist, On the equation div(|| p) + {\(\lambda\)}|u| p u = 0, Proceedings of the American Mathematical Society 109 (1990), 157–164. · Zbl 0714.35029
[8] M. Mihăilescu, P. Pucci and V. Rădulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, Comptes Rendus Mathématique. Académie des Sciences. Paris, Sér. I 345 (2007), 561–566. · Zbl 1127.35020 · doi:10.1016/j.crma.2007.10.012
[9] M. Mihăilescu, P. Pucci and V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, Journal of Mathematical Analysis and Applications 340 (2008), 687–698. · Zbl 1135.35058 · doi:10.1016/j.jmaa.2007.09.015
[10] M. Mihăilescu and V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proceedings of the American Mathematical Society 135 (2007), 2929–2937. · Zbl 1146.35067 · doi:10.1090/S0002-9939-07-08815-6
[11] M. Mihăilescu and V. Rădulescu, Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Mathematica 125 (2008), 157–167. · Zbl 1138.35070 · doi:10.1007/s00229-007-0137-8
[12] M. Mihăilescu and V. Rădulescu, Spectrum consisting in an unbounded interval for a class of nonhomogeneous differential operators, Bulletin of the London Mathematical Society 40 (2008), 972–984. · Zbl 1171.35085 · doi:10.1112/blms/bdn079
[13] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Heidelberg, 1996. · Zbl 0864.49001
[14] F. de Thélin, Sur l’espace propre associé à la première valeur propre du pseudo-laplacien, Comptes Rendus Mathématique. Académie des Sciences. Paris, Sér. I Math. 303 (1986), 355–358.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.