August  2012, 5(4): 707-714. doi: 10.3934/dcdss.2012.5.707

On some nonlocal eigenvalue problems

1. 

Department of Mathematics, Texas A&M University, Kingsville, TX 78363-8202, United States

2. 

Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901

3. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Received  February 2011 Revised  May 2011 Published  November 2011

We study a class of nonlocal eigenvalue problems related to certain boundary value problems that arise in many application areas. We construct a nondecreasing and unbounded sequence of eigenvalues that yields nontrivial critical groups for the associated variational functional using a nonstandard minimax scheme that involves the $\mathbb{Z}_2$-cohomological index. As an application we prove a multiplicity result for a class of nonlocal boundary value problems using Morse theory.
Citation: Ravi P. Agarwal, Kanishka Perera, Zhitao Zhang. On some nonlocal eigenvalue problems. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 707-714. doi: 10.3934/dcdss.2012.5.707
References:
[1]

K. C. Chang and N. Ghoussoub, The Conley index and the critical groups via an extension of Gromoll-Meyer theory, Topol. Methods Nonlinear Anal., 7 (1996), 77-93.

[2]

Kung Ching Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math., 34 (1981), 693-712. doi: 10.1002/cpa.3160340503.

[3]

Silvia Cingolani and Marco Degiovanni, Nontrivial solutions for $p$-Laplace equations with right-hand side having $p$-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203.

[4]

F. J. S. A. Corrêa and S. D. B. Menezes, Positive solutions for a class of nonlocal elliptic problems, 66 (2006), 195-206.

[5]

Francisco Júlio S. A. Corrêa and Giovany M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277. doi: 10.1017/S000497270003570X.

[6]

Francisco Júlio S. A. Corrêa and Giovany M. Figueiredo, On the existence of positive solution for an elliptic equation of Kirchhoff type via Moser iteration method, Bound. Value Probl., (2006), Art. ID 79679, 10 pp.

[7]

Francisco Júlio S. A. Corrêa and Giovany M. Figueiredo, On a $p$-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 22 (2009), 819-822.

[8]

Francisco Julio S. A. Corrêa and Rúbia G. Nascimento, On the existence of solutions of a nonlocal elliptic equation with a $p$-Kirchhoff-type term, Int. J. Math. Math. Sci., (2008), Art. ID 364085, 25 pp.

[9]

J.-N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Calculus of variations, nonsmooth analysis and related topics, Set-Valued Anal., 10 (2002), 143-164.

[10]

Weibing Deng, Zhiwen Duan, and Chunhong Xie, The blow-up rate for a degenerate parabolic equation with a non-local source, J. Math. Anal. Appl., 264 (2001), 577-597. doi: 10.1006/jmaa.2001.7696.

[11]

Weibing Deng, Yuxiang Li, and Chunhong Xie, Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations, Appl. Math. Lett., 16 (2003), 803-808. doi: 10.1016/S0893-9659(03)80118-0.

[12]

Edward R. Fadell and Paul H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174. doi: 10.1007/BF01390270.

[13]

G. Kirchhoff, "Mechanik," Teubner, Leipzig, 1883.

[14]

M. A. Krasnosel'skii, "Topological Methods in the Theory of Nonlinear Integral Equations," Translated by A. H. Armstrong, translation edited by J. Burlak, A Pergamon Press Book, The Macmillan Co., New York, 1964.

[15]

Duchao Liu, On a $p$-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302-308. doi: 10.1016/j.na.2009.06.052.

[16]

Jia Quan Liu and Shu Jie Li, An existence theorem for multiple critical points and its application, Kexue Tongbao (Chinese), 29 (1984), 1025-1027.

[17]

Kanishka Perera, Nontrivial critical groups in $p$-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal., 21 (2003), 301-309.

[18]

Kanishka Perera, Ravi P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010.

[19]

Philippe Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.

show all references

References:
[1]

K. C. Chang and N. Ghoussoub, The Conley index and the critical groups via an extension of Gromoll-Meyer theory, Topol. Methods Nonlinear Anal., 7 (1996), 77-93.

[2]

Kung Ching Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math., 34 (1981), 693-712. doi: 10.1002/cpa.3160340503.

[3]

Silvia Cingolani and Marco Degiovanni, Nontrivial solutions for $p$-Laplace equations with right-hand side having $p$-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203.

[4]

F. J. S. A. Corrêa and S. D. B. Menezes, Positive solutions for a class of nonlocal elliptic problems, 66 (2006), 195-206.

[5]

Francisco Júlio S. A. Corrêa and Giovany M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc., 74 (2006), 263-277. doi: 10.1017/S000497270003570X.

[6]

Francisco Júlio S. A. Corrêa and Giovany M. Figueiredo, On the existence of positive solution for an elliptic equation of Kirchhoff type via Moser iteration method, Bound. Value Probl., (2006), Art. ID 79679, 10 pp.

[7]

Francisco Júlio S. A. Corrêa and Giovany M. Figueiredo, On a $p$-Kirchhoff equation via Krasnoselskii's genus, Appl. Math. Lett., 22 (2009), 819-822.

[8]

Francisco Julio S. A. Corrêa and Rúbia G. Nascimento, On the existence of solutions of a nonlocal elliptic equation with a $p$-Kirchhoff-type term, Int. J. Math. Math. Sci., (2008), Art. ID 364085, 25 pp.

[9]

J.-N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals, Calculus of variations, nonsmooth analysis and related topics, Set-Valued Anal., 10 (2002), 143-164.

[10]

Weibing Deng, Zhiwen Duan, and Chunhong Xie, The blow-up rate for a degenerate parabolic equation with a non-local source, J. Math. Anal. Appl., 264 (2001), 577-597. doi: 10.1006/jmaa.2001.7696.

[11]

Weibing Deng, Yuxiang Li, and Chunhong Xie, Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations, Appl. Math. Lett., 16 (2003), 803-808. doi: 10.1016/S0893-9659(03)80118-0.

[12]

Edward R. Fadell and Paul H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174. doi: 10.1007/BF01390270.

[13]

G. Kirchhoff, "Mechanik," Teubner, Leipzig, 1883.

[14]

M. A. Krasnosel'skii, "Topological Methods in the Theory of Nonlinear Integral Equations," Translated by A. H. Armstrong, translation edited by J. Burlak, A Pergamon Press Book, The Macmillan Co., New York, 1964.

[15]

Duchao Liu, On a $p$-Kirchhoff equation via fountain theorem and dual fountain theorem, Nonlinear Anal., 72 (2010), 302-308. doi: 10.1016/j.na.2009.06.052.

[16]

Jia Quan Liu and Shu Jie Li, An existence theorem for multiple critical points and its application, Kexue Tongbao (Chinese), 29 (1984), 1025-1027.

[17]

Kanishka Perera, Nontrivial critical groups in $p$-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal., 21 (2003), 301-309.

[18]

Kanishka Perera, Ravi P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010.

[19]

Philippe Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153 (1999), 374-406. doi: 10.1006/jdeq.1998.3535.

[1]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

[2]

Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027

[3]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377

[4]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283

[5]

J. R. L. Webb, Gennaro Infante. Semi-positone nonlocal boundary value problems of arbitrary order. Communications on Pure and Applied Analysis, 2010, 9 (2) : 563-581. doi: 10.3934/cpaa.2010.9.563

[6]

Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355

[7]

Fang Li, Jerome Coville, Xuefeng Wang. On eigenvalue problems arising from nonlocal diffusion models. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 879-903. doi: 10.3934/dcds.2017036

[8]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058

[9]

John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273

[10]

Marco Squassina. Preface: Recent progresses in the theory of nonlinear nonlocal problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : i-i. doi: 10.3934/dcdss.201803i

[11]

K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546

[12]

K. Q. Lan. Multiple positive eigenvalues of conjugate boundary value problems with singularities. Conference Publications, 2003, 2003 (Special) : 501-506. doi: 10.3934/proc.2003.2003.501

[13]

Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028

[14]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061

[15]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256

[16]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003

[17]

Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507

[18]

Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819

[19]

Abdelkader Boucherif. Nonlocal problems for parabolic inclusions. Conference Publications, 2009, 2009 (Special) : 82-91. doi: 10.3934/proc.2009.2009.82

[20]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]