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March  2011, 10(2): 701-708. doi: 10.3934/cpaa.2011.10.701

An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue

Mihai Mihăilescu 1,

1. 

Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Received  May 2010 Revised  July 2010 Published  December 2010

In this paper we analyze an eigenvalue problem, involving a homogeneous Neumann boundary condition, in a smooth bounded domain. We show that the problem possesses, on the one hand, a continuous family of eigenvalues and, on the other hand, exactly one more eigenvalue which is isolated in the set of eigenvalues of the problem.
Keywords: isolated eigenvalue, homogeneous Neumann boundary condition, Eigenvalue problem, continuous family of eigenvalues..
Mathematics Subject Classification: Primary: 35D30, 35J60; Secondary: 35P3.
Citation: Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure & Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701
References:
[1]

H. Brezis, "Analyse fonctionnelle: théorie, méthodes et applications," Masson, Paris, 1992. Google Scholar

[2]

L. Gasiński and N. S. Papagiorgiu, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Chapman & Hall, 2005.  Google Scholar

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A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Birkhäuser, 2006.  Google Scholar

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M. Mihăilescu and V. Rădulescu, Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Mathematica, 125 (2008), 157-167. doi: doi:10.1007/s00229-007-0137-8.  Google Scholar

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M. Mihăilescu and V. Rădulescu, Sublinear eigenvalue problems associated to the Laplace operator revisited,, Israel Journal of Mathematics, ().   Google Scholar

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L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292. doi: doi:10.1007/BF00252910.  Google Scholar

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M. Struwe, "Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer, Heidelberg, 1996. Google Scholar

show all references

References:
[1]

H. Brezis, "Analyse fonctionnelle: théorie, méthodes et applications," Masson, Paris, 1992. Google Scholar

[2]

L. Gasiński and N. S. Papagiorgiu, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Chapman & Hall, 2005.  Google Scholar

[3]

A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Birkhäuser, 2006.  Google Scholar

[4]

M. Mihăilescu and V. Rădulescu, Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta Mathematica, 125 (2008), 157-167. doi: doi:10.1007/s00229-007-0137-8.  Google Scholar

[5]

M. Mihăilescu and V. Rădulescu, Sublinear eigenvalue problems associated to the Laplace operator revisited,, Israel Journal of Mathematics, ().   Google Scholar

[6]

L. Payne and H. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292. doi: doi:10.1007/BF00252910.  Google Scholar

[7]

M. Struwe, "Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer, Heidelberg, 1996. Google Scholar

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