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

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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

Abstract
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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 and 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.
[2]	L. Gasiński and N. S. Papagiorgiu, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Chapman & Hall, 2005.
[3]	A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Birkhäuser, 2006.
[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.
[5]	M. Mihăilescu and V. Rădulescu, Sublinear eigenvalue problems associated to the Laplace operator revisited, Israel Journal of Mathematics, in press.
[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.
[7]	M. Struwe, "Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer, Heidelberg, 1996.

show all references

References:

[1]	H. Brezis, "Analyse fonctionnelle: théorie, méthodes et applications," Masson, Paris, 1992.
[2]	L. Gasiński and N. S. Papagiorgiu, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Chapman & Hall, 2005.
[3]	A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators," Birkhäuser, 2006.
[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.
[5]	M. Mihăilescu and V. Rădulescu, Sublinear eigenvalue problems associated to the Laplace operator revisited, Israel Journal of Mathematics, in press.
[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.
[7]	M. Struwe, "Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer, Heidelberg, 1996.

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