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 & 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.   Google Scholar

[2]

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

[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.   Google Scholar

[4]

F. J. S. A. Corrêa and S. D. B. Menezes, Positive solutions for a class of nonlocal elliptic problems,, \textbf{66} (2006), 66 (2006), 195.   Google Scholar

[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.  doi: 10.1017/S000497270003570X.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[9]

J.-N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals,, Calculus of variations, 10 (2002), 143.   Google Scholar

[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.  doi: 10.1006/jmaa.2001.7696.  Google Scholar

[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.  doi: 10.1016/S0893-9659(03)80118-0.  Google Scholar

[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.  doi: 10.1007/BF01390270.  Google Scholar

[13]

G. Kirchhoff, "Mechanik,", Teubner, (1883).   Google Scholar

[14]

M. A. Krasnosel'skii, "Topological Methods in the Theory of Nonlinear Integral Equations,", Translated by A. H. Armstrong, (1964).   Google Scholar

[15]

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

[16]

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

[17]

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

[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, (2010).   Google Scholar

[19]

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

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.   Google Scholar

[2]

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

[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.   Google Scholar

[4]

F. J. S. A. Corrêa and S. D. B. Menezes, Positive solutions for a class of nonlocal elliptic problems,, \textbf{66} (2006), 66 (2006), 195.   Google Scholar

[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.  doi: 10.1017/S000497270003570X.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[9]

J.-N. Corvellec and A. Hantoute, Homotopical stability of isolated critical points of continuous functionals,, Calculus of variations, 10 (2002), 143.   Google Scholar

[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.  doi: 10.1006/jmaa.2001.7696.  Google Scholar

[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.  doi: 10.1016/S0893-9659(03)80118-0.  Google Scholar

[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.  doi: 10.1007/BF01390270.  Google Scholar

[13]

G. Kirchhoff, "Mechanik,", Teubner, (1883).   Google Scholar

[14]

M. A. Krasnosel'skii, "Topological Methods in the Theory of Nonlinear Integral Equations,", Translated by A. H. Armstrong, (1964).   Google Scholar

[15]

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

[16]

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

[17]

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

[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, (2010).   Google Scholar

[19]

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

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