
-
Previous Article
Mild solutions to the time fractional Navier-Stokes delay differential inclusions
- DCDS-B Home
- This Issue
-
Next Article
Pollution control for switching diffusion models: Approximation methods and numerical results
Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient
1. | Dpto. de Matemática, Campus Universitário Darcy Ribeiro, Universidade de Brasília, 70910-900, Brasília - DF, Brazil |
2. | Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, Sevilla, C/. Tarfia s/n, 41012, Spain |
In this paper we study a stationary problem arising from population dynamics with a local and nonlocal variable diffusion coefficient. We show the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global structure of this continuum depends on the value of the nonlocal diffusion at infinity and the relative position of the refuge of the species and of the sets where it diffuses locally and not locally, respectively.
References:
[1] |
C. O. Alves, F. J. S. A. Corrêa and M. Chipot,
On a class of intermediate local-nonlocal elliptic problems, Topol. Methods Nonlinear Anal., 49 (2017), 497-509.
|
[2] |
A. Ambrosetti and J. L. Gámez,
Branches of positive solutions for some semilinear Schrödinger equations, Math. Z.,, 224 (1997), 347-362.
doi: 10.1007/PL00004586. |
[3] |
D. Arcoya, T. Leonori and A. Primo,
Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano Theorem, Acta Appl. Math.,, 127 (2013), 87-104.
doi: 10.1007/s10440-012-9792-1. |
[4] |
H. Brezis and L. Oswald,
Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.
doi: 10.1016/0362-546X(86)90011-8. |
[5] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003.
doi: 10.1002/0470871296. |
[6] |
T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio,
Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal., 121 (2015), 3-18.
doi: 10.1016/j.na.2014.07.011. |
[7] |
M. Chipot and F. J. S. A. Corrêa,
Boundary layer solutions to functional elliptic equations, Bull. Braz. Math. Soc., New Series, 40 (2009), 1-13.
doi: 10.1007/s00574-009-0017-9. |
[8] |
M. Chipot and P. Roy,
Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.
|
[9] |
T. S. Figueiredo-Sousa, C. Morales-Rodrigo and A. Suárez, A non-local non-autonomous diffusion problem: linear and sublinear cases, Z. Angew. Math. Phys., 68 (2017), Art. 108, 20 pp.
doi: 10.1007/s00033-017-0856-y. |
[10] |
T. S. Figueiredo-Sousa, C. Morales-Rodrigo and A. Suárez, The influence of a metasolution on the behaviour of the logistic equation with nonlocal diffusion coefficient, Calc. Var. Partial Differential Equations, 57 (2018), Art. 100, 26 pp.
doi: 10.1007/s00526-018-1385-z. |
[11] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8664. |
[12] |
J. López-Gómez,
Metasolutions of Parabolic Equations in Population Dynamics, Taylor and Francis Group, 2016. |
[13] |
A. Molino and J. D. Rossi, A concave-convex problem with a variable operator, Calc. Var. Partial Differential Equations, 57 (2018), Art. 10, 26 pp.
doi: 10.1007/s00526-017-1291-9. |
[14] |
T. Ouyang,
On the positive solutions of semilinear equations $\Delta u + {\rm{ }}\lambda u - h{u^p} = 0$ on the compact manifolds, Transactions of the American Mathematical Society, 331 (1992), 503-527.
doi: 10.2307/2154124. |
[15] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[16] |
P. Roy,
Existence results for some nonlocal problems, Differ. Equ. Appl., 6 (2014), 361-381.
doi: 10.7153/dea-06-20. |
[17] |
B. Yan and T. Ma, The existence and multiplicity of positive solutions for a class of nonlocal elliptic problem, Bound. Value. Probl., 2016, Paper No. 165, 35 pp.
doi: 10.1186/s13661-016-0670-z. |
[18] |
B. Yan and D. Wang,
The multiplicity of positive solutions for a class of nonlocal elliptic problem, J. Math. Anal. Appl., 442 (2016), 72-102.
doi: 10.1016/j.jmaa.2016.04.023. |
show all references
To Peter Kloeden for his 70th birthday
References:
[1] |
C. O. Alves, F. J. S. A. Corrêa and M. Chipot,
On a class of intermediate local-nonlocal elliptic problems, Topol. Methods Nonlinear Anal., 49 (2017), 497-509.
|
[2] |
A. Ambrosetti and J. L. Gámez,
Branches of positive solutions for some semilinear Schrödinger equations, Math. Z.,, 224 (1997), 347-362.
doi: 10.1007/PL00004586. |
[3] |
D. Arcoya, T. Leonori and A. Primo,
Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano Theorem, Acta Appl. Math.,, 127 (2013), 87-104.
doi: 10.1007/s10440-012-9792-1. |
[4] |
H. Brezis and L. Oswald,
Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.
doi: 10.1016/0362-546X(86)90011-8. |
[5] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003.
doi: 10.1002/0470871296. |
[6] |
T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio,
Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal., 121 (2015), 3-18.
doi: 10.1016/j.na.2014.07.011. |
[7] |
M. Chipot and F. J. S. A. Corrêa,
Boundary layer solutions to functional elliptic equations, Bull. Braz. Math. Soc., New Series, 40 (2009), 1-13.
doi: 10.1007/s00574-009-0017-9. |
[8] |
M. Chipot and P. Roy,
Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.
|
[9] |
T. S. Figueiredo-Sousa, C. Morales-Rodrigo and A. Suárez, A non-local non-autonomous diffusion problem: linear and sublinear cases, Z. Angew. Math. Phys., 68 (2017), Art. 108, 20 pp.
doi: 10.1007/s00033-017-0856-y. |
[10] |
T. S. Figueiredo-Sousa, C. Morales-Rodrigo and A. Suárez, The influence of a metasolution on the behaviour of the logistic equation with nonlocal diffusion coefficient, Calc. Var. Partial Differential Equations, 57 (2018), Art. 100, 26 pp.
doi: 10.1007/s00526-018-1385-z. |
[11] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8664. |
[12] |
J. López-Gómez,
Metasolutions of Parabolic Equations in Population Dynamics, Taylor and Francis Group, 2016. |
[13] |
A. Molino and J. D. Rossi, A concave-convex problem with a variable operator, Calc. Var. Partial Differential Equations, 57 (2018), Art. 10, 26 pp.
doi: 10.1007/s00526-017-1291-9. |
[14] |
T. Ouyang,
On the positive solutions of semilinear equations $\Delta u + {\rm{ }}\lambda u - h{u^p} = 0$ on the compact manifolds, Transactions of the American Mathematical Society, 331 (1992), 503-527.
doi: 10.2307/2154124. |
[15] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[16] |
P. Roy,
Existence results for some nonlocal problems, Differ. Equ. Appl., 6 (2014), 361-381.
doi: 10.7153/dea-06-20. |
[17] |
B. Yan and T. Ma, The existence and multiplicity of positive solutions for a class of nonlocal elliptic problem, Bound. Value. Probl., 2016, Paper No. 165, 35 pp.
doi: 10.1186/s13661-016-0670-z. |
[18] |
B. Yan and D. Wang,
The multiplicity of positive solutions for a class of nonlocal elliptic problem, J. Math. Anal. Appl., 442 (2016), 72-102.
doi: 10.1016/j.jmaa.2016.04.023. |


[1] |
Qinghua Luo. Damped Klein-Gordon equation with variable diffusion coefficient. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3959-3974. doi: 10.3934/cpaa.2021139 |
[2] |
Armel Ovono Andami. From local to nonlocal in a diffusion model. Conference Publications, 2011, 2011 (Special) : 54-60. doi: 10.3934/proc.2011.2011.54 |
[3] |
Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228 |
[4] |
Masaru Ikehata, Yavar Kian. The enclosure method for the detection of variable order in fractional diffusion equations. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022036 |
[5] |
Aníbal Rodríguez-Bernal, Silvia Sastre-Gómez. Nonlinear nonlocal reaction-diffusion problem with local reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1731-1765. doi: 10.3934/dcds.2021170 |
[6] |
Guangying Lv, Hongjun Gao, Jinlong Wei, Jiang-Lun Wu. On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022119 |
[7] |
Rafael de la Rosa, María Santos Bruzón. Differential invariants of a generalized variable-coefficient Gardner equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 747-757. doi: 10.3934/dcdss.2018047 |
[8] |
Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319 |
[9] |
Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 |
[10] |
Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226 |
[11] |
Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173 |
[12] |
Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5105-5139. doi: 10.3934/dcds.2021070 |
[13] |
Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041 |
[14] |
Bastian Harrach. Simultaneous determination of the diffusion and absorption coefficient from boundary data. Inverse Problems and Imaging, 2012, 6 (4) : 663-679. doi: 10.3934/ipi.2012.6.663 |
[15] |
Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems and Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285 |
[16] |
Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control and Related Fields, 2021, 11 (4) : 965-985. doi: 10.3934/mcrf.2020054 |
[17] |
André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 2022, 11 (3) : 749-779. doi: 10.3934/eect.2021024 |
[18] |
Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301 |
[19] |
J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure and Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037 |
[20] |
Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks and Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]