July  2016, 36(7): 4027-4049. doi: 10.3934/dcds.2016.36.4027

Principal eigenvalues for some nonlocal eigenvalue problems and applications

1. 

School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China, China

Received  April 2015 Revised  November 2015 Published  March 2016

This paper is concerned with the principal eigenvalues of some nonlocal operators. We first derive a result on the limit of certain sequences of principal eigenvalues associated with some nonlocal eigenvalue problems. Then, such a result is used to study the existence, uniqueness and asymptotic behavior of positive solutions to a nonlocal stationary problem with a parameter. Finally, the long-time behavior of the solutions of the corresponding nonlocal evolution equation and the asymptotic behavior of positive stationary solutions on parameter are discussed.
Citation: Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
References:
[1]

AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037.  Google Scholar

[3]

J. Statist. Phys., 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625.  Google Scholar

[4]

J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[5]

J. Funct. Anal., 40 (1981), 1-29. doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

[6]

Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105.  Google Scholar

[7]

J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.  Google Scholar

[8]

J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[9]

Arch. Ration. Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8.  Google Scholar

[10]

Israel J. Math., 170 (2009), 53-60. doi: 10.1007/s11856-009-0019-8.  Google Scholar

[11]

SIAM J. Math. Anal., 39 (2008), 1693-1709. doi: 10.1137/060676854.  Google Scholar

[12]

J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[13]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223. doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[14]

Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446. doi: 10.3934/dcds.2015.35.1421.  Google Scholar

[15]

Ser. Math. Comput. Biol., John Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.  Google Scholar

[16]

SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.  Google Scholar

[17]

J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar

[18]

in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.  Google Scholar

[19]

Arch. Ration. Mech. Anal., 145 (1998), 261-289. doi: 10.1007/s002050050130.  Google Scholar

[20]

Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037.  Google Scholar

[21]

J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015.  Google Scholar

[22]

Differential Integral Equations, 18 (2005), 1299-1320.  Google Scholar

[23]

J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1.  Google Scholar

[24]

J. Math. Pures Appl., 92 (2009), 163-187. doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[25]

SIAM J. Appl. Math., 71 (2011), 1241-1262. doi: 10.1137/100788033.  Google Scholar

[26]

J. Math. Biol., 65 (2012), 403-439. doi: 10.1007/s00285-011-0465-x.  Google Scholar

[27]

Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.  Google Scholar

[28]

Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[29]

J. Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.  Google Scholar

[30]

J. Differential Equations, 148 (1998), 47-64. doi: 10.1006/jdeq.1998.3456.  Google Scholar

[31]

Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124.  Google Scholar

[32]

Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y.  Google Scholar

[33]

Nonlinear Anal., 74 (2011), 3501-3509. doi: 10.1016/j.na.2011.02.034.  Google Scholar

[34]

J. Differential Equations, 257 (2014), 1372-1402. doi: 10.1016/j.jde.2014.05.005.  Google Scholar

[35]

Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238. doi: 10.3934/dcds.2015.35.3217.  Google Scholar

[36]

J. Differential Equations, 251 (2011), 551-581. doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[37]

Nonlinear Anal., 74 (2011), 814-826. doi: 10.1016/j.na.2010.09.032.  Google Scholar

[38]

J. Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[39]

Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[40]

Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[41]

Nonlinear Anal., 72 (2010), 4466-4474. doi: 10.1016/j.na.2010.02.021.  Google Scholar

show all references

References:
[1]

AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037.  Google Scholar

[3]

J. Statist. Phys., 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625.  Google Scholar

[4]

J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[5]

J. Funct. Anal., 40 (1981), 1-29. doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

[6]

Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105.  Google Scholar

[7]

J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.  Google Scholar

[8]

J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[9]

Arch. Ration. Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8.  Google Scholar

[10]

Israel J. Math., 170 (2009), 53-60. doi: 10.1007/s11856-009-0019-8.  Google Scholar

[11]

SIAM J. Math. Anal., 39 (2008), 1693-1709. doi: 10.1137/060676854.  Google Scholar

[12]

J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[13]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223. doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[14]

Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446. doi: 10.3934/dcds.2015.35.1421.  Google Scholar

[15]

Ser. Math. Comput. Biol., John Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.  Google Scholar

[16]

SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.  Google Scholar

[17]

J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar

[18]

in Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.  Google Scholar

[19]

Arch. Ration. Mech. Anal., 145 (1998), 261-289. doi: 10.1007/s002050050130.  Google Scholar

[20]

Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037.  Google Scholar

[21]

J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015.  Google Scholar

[22]

Differential Integral Equations, 18 (2005), 1299-1320.  Google Scholar

[23]

J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1.  Google Scholar

[24]

J. Math. Pures Appl., 92 (2009), 163-187. doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[25]

SIAM J. Appl. Math., 71 (2011), 1241-1262. doi: 10.1137/100788033.  Google Scholar

[26]

J. Math. Biol., 65 (2012), 403-439. doi: 10.1007/s00285-011-0465-x.  Google Scholar

[27]

Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.  Google Scholar

[28]

Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[29]

J. Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.  Google Scholar

[30]

J. Differential Equations, 148 (1998), 47-64. doi: 10.1006/jdeq.1998.3456.  Google Scholar

[31]

Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124.  Google Scholar

[32]

Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y.  Google Scholar

[33]

Nonlinear Anal., 74 (2011), 3501-3509. doi: 10.1016/j.na.2011.02.034.  Google Scholar

[34]

J. Differential Equations, 257 (2014), 1372-1402. doi: 10.1016/j.jde.2014.05.005.  Google Scholar

[35]

Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238. doi: 10.3934/dcds.2015.35.3217.  Google Scholar

[36]

J. Differential Equations, 251 (2011), 551-581. doi: 10.1016/j.jde.2011.04.020.  Google Scholar

[37]

Nonlinear Anal., 74 (2011), 814-826. doi: 10.1016/j.na.2010.09.032.  Google Scholar

[38]

J. Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[39]

Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[40]

Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[41]

Nonlinear Anal., 72 (2010), 4466-4474. doi: 10.1016/j.na.2010.02.021.  Google Scholar

[1]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[2]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

[3]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[4]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[5]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[6]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[7]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[8]

Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021103

[9]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029

[10]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[11]

Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021064

[12]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065

[13]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[14]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270

[15]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[16]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[17]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019

[18]

Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061

[19]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[20]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (142)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]