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

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.

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 and Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311
References:
[1]

C. O. AlvesF. 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. ArcoyaT. 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. CaraballoM. 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. AlvesF. 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. ArcoyaT. 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. CaraballoM. 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.

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