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

Previous Article
Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay
CPAA Home
This Issue
Next Article
Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents

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

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

Received May 2010 Revised July 2010 Published December 2010

Abstract
Related Papers

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, (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,", Birkh\, (2006). Google Scholar
[4]	M. Mihăilescu and V. Rădulescu, Continuous spectrum for a class of nonhomogeneous differential operators,, Manuscripta Mathematica, 125 (2008), 157. 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. doi: doi:10.1007/BF00252910. Google Scholar
[7]	M. Struwe, "Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer, (1996). Google Scholar

show all references

References:

[1]	H. Brezis, "Analyse fonctionnelle: théorie, méthodes et applications,", Masson, (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,", Birkh\, (2006). Google Scholar
[4]	M. Mihăilescu and V. Rădulescu, Continuous spectrum for a class of nonhomogeneous differential operators,, Manuscripta Mathematica, 125 (2008), 157. 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. doi: doi:10.1007/BF00252910. Google Scholar
[7]	M. Struwe, "Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer, (1996). Google Scholar

[1]	Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[2]	Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
[3]	VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003
[4]	Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1
[5]	Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
[6]	Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491
[7]	Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[8]	Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201
[9]	Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335
[10]	J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[11]	Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[12]	Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[13]	Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016
[14]	David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[15]	Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[16]	Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343
[17]	Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[18]	Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[19]	Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5217-5252. doi: 10.3934/dcds.2013.33.5217
[20]	Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences