American Institute of Mathematical Sciences

Advanced
August  2019, 24(8): 3689-3711. doi: 10.3934/dcdsb.2018311

Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient

Giovany M. Figueiredo 1, , Tarcyana S. Figueiredo-Sousa 2, , Cristian Morales-Rodrigo 2, and  Antonio Suárez 2,

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

To Peter Kloeden for his 70th birthday

Received  March 2018 Revised  July 2018 Published  August 2019 Early access  October 2018

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.

Keywords: Bifurcation method, variable local and nonlocal diffusion coefficient.
Mathematics Subject Classification: Primary: 35B09, 35B32; Secondary: 35J60, 35P30.
Citation: Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Chipot and P. Roy, Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.  Google Scholar

[12]

J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, Taylor and Francis Group, 2016.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Roy, Existence results for some nonlocal problems, Differ. Equ. Appl., 6 (2014), 361-381.  doi: 10.7153/dea-06-20.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Chipot and P. Roy, Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.  Google Scholar

[12]

J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, Taylor and Francis Group, 2016.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Roy, Existence results for some nonlocal problems, Differ. Equ. Appl., 6 (2014), 361-381.  doi: 10.7153/dea-06-20.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Bifurcation diagrams when $\lambda _0<\lambda _\infty<\infty$ and $\lambda _\infty<\lambda _0<\infty$, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Bifurcation diagrams when $\lambda _\infty = 0$ and $\lambda_\infty = \infty$, respectively. For example, this last diagram appears when $b\geq b_0>0$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Qinghua Luo. Damped Klein-Gordon equation with variable diffusion coefficient. Communications on Pure & Applied Analysis, , () : -. 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 & Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228
[4]

Rafael de la Rosa, María Santos Bruzón. Differential invariants of a generalized variable-coefficient Gardner equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 747-757. doi: 10.3934/dcdss.2018047
[5]

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 & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319
[6]

Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226
[7]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[8]

Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173
[9]

Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5105-5139. doi: 10.3934/dcds.2021070
[10]

Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
[11]

Bastian Harrach. Simultaneous determination of the diffusion and absorption coefficient from boundary data. Inverse Problems & Imaging, 2012, 6 (4) : 663-679. doi: 10.3934/ipi.2012.6.663
[12]

Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems & Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285
[13]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020054
[14]

André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021024
[15]

Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301
[16]

J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037
[17]

Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014
[18]

Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
[19]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495
[20]

Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021156

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (212)
  • HTML views (746)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences