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doi: 10.3934/dcdss.2022018
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Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations

Instituto Matemático Interdisplinar, Depto. Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

* Corresponding author: Jesús Ildefonso Díaz

Dedicated to Georg Hetzer: Elegant mathematician and very good friend in his 75th

Received  October 2021 Early access February 2022

Fund Project: Partially supported the UCM Research Group MOMAT (ref. 910480) and the projects MTM2017-85449-P and PID2020-112517GB-I00 of the DGISPI, Spain

We establish the existence of bounded very weak solutions to a large class of stationary diffusive logistic equations with weights by constructing suitable sub and supersolutions. This class of problems corresponds to the case in which the absorption term dominates over the forcing term. The case of simultaneous singular nonlinearities and singular weights is also considered. This shows that if limitations in the growth of a population are distributed and unbounded, but satisfy some mild integrability assumption in terms of the distance to the boundary, solutions can still be bounded. The results extend several papers in the literature.

Citation: Jesús Ildefonso Díaz, Jesús Hernández. Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022018
References:
[1]

M. Bertsch and R. Rostamian, The principle of linearized stability for a class of degenerate diffusion equations, J. Differ. Equat., 57 (1985), 373-405.  doi: 10.1016/0022-0396(85)90062-2.

[2]

L. Boccardo and L. Orsina, Sublinear elliptic equations in $L^{s}$, Houston J. Math., 20 (1994), 99–114.

[3]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363–380. doi: 10.1007/s00526-009-0266-x.

[4]

H. Brezis and F. E. Browder, Some properties of higher order Sobolev spaces, J. Math. Pures Appl., 61 (1982), 245–259.

[5]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 223–262.

[6]

H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_{t}-\Delta u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73–90.

[7]

H. Brezis and S. Kamin, Sublinear elliptic equations in ${\bf{R}}^n$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[8]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137–151.

[9]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10(1986), 55–64. doi: 10.1016/0362-546X(86)90011-8.

[10]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Chichester, UK 2003. doi: 10.1002/0470871296.

[11]

M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with singular nonlinearity, Comm. PDEs, 2(1977), 193–222. doi: 10.1080/03605307708820029.

[12]

A. N. Dao and J. I. Díaz, The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption, J. Differential Equations, 263 (2017), 6764–6804. doi: 10.1016/j.jde.2017.07.029.

[13]

D. G. de Figueiredo, Positive Solutions of Semilinear Elliptic Problems, Lectures Notes, Sao Paulo, 1981.

[14]

J. I. Díaz, New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems, Pure Appl. Funct. Anal., 5 (2020), 925–949.

[15]

J. I. Díaz and J. Hernández, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption: An overview and open problems, In Proceedings of the Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems, (2012). Electronic Journal of Differential Equations, Conference, 21 (2014), 31–44.

[16]

J. I. Díaz and J. Hernández, Linearized stability for degenerate and singular semilinear and quasilinear parabolic problems: The linearized singular equations, Topol. Methods Nonlinear Anal., 54 (2019), 937–966. doi: 10.12775/tmna.2019.091.

[17]

J. I. Díaz, J. Hernández and Y. Sh. Ilyasov, On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets, Adv. Nonlinear Anal., 9(2020), 1046–1065. doi: 10.1515/anona-2020-0030.

[18]

J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Maths., 79 (2011), 233–245. doi: 10.1007/s00032-011-0151-x.

[19]

J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak non-negative solutions to some second order semilinear elliptic problems with a singular absorption term, In preparation.

[20]

J. I. Díaz and S. Kamin, Convergence to travelling waves for quasilinear Fisher-KPP type equations, J. Math. Anal. Appl., 390 (2012), 74–85. doi: 10.1016/j.jmaa.2012.01.018.

[21]

J. I. Díaz, J.-M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations, 12(1987), 1333–1344. doi: 10.1080/03605308708820531.

[22]

J. I. Díaz and J. M. Rakotoson, On very weak solutions of semilinear elliptic equations with right hand side data integrable with respect to the distance to the boundary, Discrete Contin. Dyn. Syst., 27 (2010), 1037–1058. doi: 10.3934/dcds.2010.27.1037.

[23]

J. I. Díaz and J. M. Rakotoson, $L^{1}(\Omega; dist(x; \partial \Omega))$-problems and their applications revisited, Electron. J. Diff. Eqns, Conference, 21 (2014), 45–59.

[24]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman and Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.

[25]

N. El Berdan, J. I. Díaz and J. M. Rakotoson, The uniform Hopf Inequality for discontinuous coefficients and optimal regularity in BMO for singular problems, J. Math. Anal. Appl., 437 (2016), 350–379. doi: 10.1016/j.jmaa.2015.11.065.

[26] M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford University Press, 2008. 
[27]

J. Giacomoni, P. Sauvy and S. Shmarev, Complete quenching for a quasilinear parabolic equation, J. Math. Anal. Appl., 410(2014), 607–624. doi: 10.1016/j.jmaa.2013.08.051.

[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlín, 2d ed., 1983. doi: 10.1007/978-3-642-61798-0.

[29]

J. Hernández, Qualitative methods for nonlinear diffusion equations, in Nonlinear Diffusion Problems, A. Fasano and M. Primicerio (eds.), Springer Lectures Notes, 1224 (1986), 47–118. doi: 10.1007/BFb0072686.

[30]

J. Hernández, Positive solutions for the logistic equation with unbounded weights, In Reaction Diffusion Systems (Trieste, 1995), G. Caristi and E. Mitidieri (eds.), Marcel Dekker. New York, 1998,183–197.

[31]

J. HernándezF. J. Mancebo and J. M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Lin., 19 (2002), 777-813.  doi: 10.1016/s0294-1449(02)00102-6.

[32]

J. Hernández, F. J. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 41–62. doi: 10.1017/S030821050500065X.

[33]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999–1030. doi: 10.1080/03605308008820162.

[34]

G. Hetzer, W. Shen and Sh. Zhu, Asymptotic behavior of positive solutions of random and stochastic parabolic equations of fisher and Kolmogorov types, J. Dynam. Differential Equations, 14 (2002), 139–188. doi: 10.1023/A:1012932212645.

[35]

U. Kaufmann, H. Ramos Quoirin and K. Umezu, Past and recent contributions to indefinite sublinear elliptic problems, Rend. Istit. Mat. Univ. Trieste, 52 (2020), 217–241. doi: 10.1080/0305215x.2019.1577412.

[36]

A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, New-York, 1985.

[37]

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in ${\bf{R}}^n$, Trans. Amer. Math. Soc., 350 (1998), 3237–3255. doi: 10.1090/S0002-9947-98-02122-9.

[38]

M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429–2438. doi: 10.1090/S0002-9939-08-09231-9.

[39] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[40]

J.-M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications, Adv. Differential Equations, 16 (2011), 867–894.

[41]

Ph. Souplet, Optimal regularity conditions for elliptic problems via $L_{\delta }^{p}$-spaces, Duke Math. J., 127 (2005), 175–192. doi: 10.1215/S0012-7094-04-12715-0.

show all references

References:
[1]

M. Bertsch and R. Rostamian, The principle of linearized stability for a class of degenerate diffusion equations, J. Differ. Equat., 57 (1985), 373-405.  doi: 10.1016/0022-0396(85)90062-2.

[2]

L. Boccardo and L. Orsina, Sublinear elliptic equations in $L^{s}$, Houston J. Math., 20 (1994), 99–114.

[3]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363–380. doi: 10.1007/s00526-009-0266-x.

[4]

H. Brezis and F. E. Browder, Some properties of higher order Sobolev spaces, J. Math. Pures Appl., 61 (1982), 245–259.

[5]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 223–262.

[6]

H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_{t}-\Delta u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73–90.

[7]

H. Brezis and S. Kamin, Sublinear elliptic equations in ${\bf{R}}^n$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[8]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137–151.

[9]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10(1986), 55–64. doi: 10.1016/0362-546X(86)90011-8.

[10]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Chichester, UK 2003. doi: 10.1002/0470871296.

[11]

M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with singular nonlinearity, Comm. PDEs, 2(1977), 193–222. doi: 10.1080/03605307708820029.

[12]

A. N. Dao and J. I. Díaz, The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption, J. Differential Equations, 263 (2017), 6764–6804. doi: 10.1016/j.jde.2017.07.029.

[13]

D. G. de Figueiredo, Positive Solutions of Semilinear Elliptic Problems, Lectures Notes, Sao Paulo, 1981.

[14]

J. I. Díaz, New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems, Pure Appl. Funct. Anal., 5 (2020), 925–949.

[15]

J. I. Díaz and J. Hernández, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption: An overview and open problems, In Proceedings of the Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems, (2012). Electronic Journal of Differential Equations, Conference, 21 (2014), 31–44.

[16]

J. I. Díaz and J. Hernández, Linearized stability for degenerate and singular semilinear and quasilinear parabolic problems: The linearized singular equations, Topol. Methods Nonlinear Anal., 54 (2019), 937–966. doi: 10.12775/tmna.2019.091.

[17]

J. I. Díaz, J. Hernández and Y. Sh. Ilyasov, On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets, Adv. Nonlinear Anal., 9(2020), 1046–1065. doi: 10.1515/anona-2020-0030.

[18]

J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Maths., 79 (2011), 233–245. doi: 10.1007/s00032-011-0151-x.

[19]

J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak non-negative solutions to some second order semilinear elliptic problems with a singular absorption term, In preparation.

[20]

J. I. Díaz and S. Kamin, Convergence to travelling waves for quasilinear Fisher-KPP type equations, J. Math. Anal. Appl., 390 (2012), 74–85. doi: 10.1016/j.jmaa.2012.01.018.

[21]

J. I. Díaz, J.-M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations, 12(1987), 1333–1344. doi: 10.1080/03605308708820531.

[22]

J. I. Díaz and J. M. Rakotoson, On very weak solutions of semilinear elliptic equations with right hand side data integrable with respect to the distance to the boundary, Discrete Contin. Dyn. Syst., 27 (2010), 1037–1058. doi: 10.3934/dcds.2010.27.1037.

[23]

J. I. Díaz and J. M. Rakotoson, $L^{1}(\Omega; dist(x; \partial \Omega))$-problems and their applications revisited, Electron. J. Diff. Eqns, Conference, 21 (2014), 45–59.

[24]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman and Hall/CRC, Boca Raton, FL, 2011. doi: 10.1201/b10802.

[25]

N. El Berdan, J. I. Díaz and J. M. Rakotoson, The uniform Hopf Inequality for discontinuous coefficients and optimal regularity in BMO for singular problems, J. Math. Anal. Appl., 437 (2016), 350–379. doi: 10.1016/j.jmaa.2015.11.065.

[26] M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford University Press, 2008. 
[27]

J. Giacomoni, P. Sauvy and S. Shmarev, Complete quenching for a quasilinear parabolic equation, J. Math. Anal. Appl., 410(2014), 607–624. doi: 10.1016/j.jmaa.2013.08.051.

[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlín, 2d ed., 1983. doi: 10.1007/978-3-642-61798-0.

[29]

J. Hernández, Qualitative methods for nonlinear diffusion equations, in Nonlinear Diffusion Problems, A. Fasano and M. Primicerio (eds.), Springer Lectures Notes, 1224 (1986), 47–118. doi: 10.1007/BFb0072686.

[30]

J. Hernández, Positive solutions for the logistic equation with unbounded weights, In Reaction Diffusion Systems (Trieste, 1995), G. Caristi and E. Mitidieri (eds.), Marcel Dekker. New York, 1998,183–197.

[31]

J. HernándezF. J. Mancebo and J. M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Lin., 19 (2002), 777-813.  doi: 10.1016/s0294-1449(02)00102-6.

[32]

J. Hernández, F. J. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 41–62. doi: 10.1017/S030821050500065X.

[33]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999–1030. doi: 10.1080/03605308008820162.

[34]

G. Hetzer, W. Shen and Sh. Zhu, Asymptotic behavior of positive solutions of random and stochastic parabolic equations of fisher and Kolmogorov types, J. Dynam. Differential Equations, 14 (2002), 139–188. doi: 10.1023/A:1012932212645.

[35]

U. Kaufmann, H. Ramos Quoirin and K. Umezu, Past and recent contributions to indefinite sublinear elliptic problems, Rend. Istit. Mat. Univ. Trieste, 52 (2020), 217–241. doi: 10.1080/0305215x.2019.1577412.

[36]

A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, New-York, 1985.

[37]

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in ${\bf{R}}^n$, Trans. Amer. Math. Soc., 350 (1998), 3237–3255. doi: 10.1090/S0002-9947-98-02122-9.

[38]

M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429–2438. doi: 10.1090/S0002-9939-08-09231-9.

[39] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. 
[40]

J.-M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications, Adv. Differential Equations, 16 (2011), 867–894.

[41]

Ph. Souplet, Optimal regularity conditions for elliptic problems via $L_{\delta }^{p}$-spaces, Duke Math. J., 127 (2005), 175–192. doi: 10.1215/S0012-7094-04-12715-0.

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