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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  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

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