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April  2019, 12(2): 171-188. doi: 10.3934/dcdss.2019012

Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$

Imed Bachar 1,, and  Habib Mâagli 2,

1. 

King Saud University, College of Science, Mathematics Department, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2. 

King Abdulaziz University, College of Sciences and Arts, Rabigh Campus, Department of Mathematics, P.O. Box 344, Rabigh 21911, Saudi Arabia

* Corresponding author: Imed Bachar

Dedicated to Vicenţiu D. Rǎdulescu on his sixtieth birthday

Received  May 2017 Revised  December 2017 Published  August 2018

We consider the following singular semilinear problem
$\left\{ \begin{array}{l} - \Delta u(x) = a(x){u^\sigma }(x),{\rm{ }}x \in \Omega \backslash \{ 0\} ({\rm{in\;the\;distributional\;sense}}),\\\;u > 0,{\rm{ on}}\;\Omega \backslash \{ 0\} ,\\\mathop {\lim }\limits_{\left| x \right| \to 0} \frac{{u(x)}}{{\ln \left| x \right|}} = 0,\\u(x) = 0,\;x \in \partial \Omega ,\end{array} \right.$
where
$σ <1,$
$Ω $
 is a bounded regular domain in
$\mathbb{R}^{2}$
 with
$0∈ Ω .$
 The weight function
$a(x)$
 is requiredto be positive and continuous in
$Ω \backslash \{0\}$
 with thepossibility to be singular at
$x = 0$
 and/or at the boundary
$\partial Ω. $
 When the function
$a$
 satisfies sharp estimates related to Karamataclass, we prove the existence and global asymptotic behavior of a positivecontinuous solution on
$\overline{Ω }\backslash \{0\}$
 which couldblow-up at
$0$
.
Keywords: Singular positive solution, Green's function, Karamata class, Kato class, blow-up.
Mathematics Subject Classification: Primary: 34B16, 34B27; Secondary: 34B18.
Citation: Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012
References:
[1]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl., vol. 27, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[2]

J. Bliedtner and W. Hansen, Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, 1986. doi: 10.1007/978-3-642-71131-2.  Google Scholar

[3]

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

[4]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, Third edition. Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.  Google Scholar

[5]

R. Chemmam, H. Mâagli, S. Masmoudi and M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal., 1 (2012), 301-318.  doi: 10.1515/anona-2012-0008.  Google Scholar

[6] K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer-Verlag, 1995.  doi: 10.1007/978-3-642-57856-4.  Google Scholar
[7]

F. Cirstea and V. D. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Math. Acad. Sci. Paris., 335 (2002), 447-452.  doi: 10.1016/S1631-073X(02)02503-7.  Google Scholar

[8]

F. Cirstea and V. D. Rădulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Transactions Amer. Math. Soc., 359 (2007), 3275-3286.  doi: 10.1090/S0002-9947-07-04107-4.  Google Scholar

[9]

M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar

[10]

S. Dumont, L. Dupaigne, O. Goubet and V. D. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.  doi: 10.1515/ans-2007-0205.  Google Scholar

[11]

M. Ghergu and V. D. Rădulescu, PDEs Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer Verlag, Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.  Google Scholar

[12]

M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Applications, Vol. 37, Oxford University Press, 2008.  Google Scholar

[13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, third ed., Springer Verlag, Berlin, 1983.  doi: 10.1007/978-3-642-61798-0.  Google Scholar
[14]

J. Karamata, Sur un mode de croissance régulière. Thé orèmes fondamentaux, Bull. Soc. Math. France., 61 (1933), 55-62.   Google Scholar

[15]

S. Karntz and S. Stević, On the iterated logarithmic Bloch space on the unit ball, Nonlinear Anal. TMA., 71 (2009), 1772-1795.  doi: 10.1016/j.na.2009.01.013.  Google Scholar

[16]

A. C. Lazer and P. J. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.  doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar

[17]

S. Li and S. Stević, On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces, Appl. Math. Comput., 215 (2009), 3106-3115.  doi: 10.1016/j.amc.2009.10.004.  Google Scholar

[18]

H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal., 74 (2011), 2941-2947.  doi: 10.1016/j.na.2011.01.011.  Google Scholar

[19]

H. Mâagli and L. Mâatoug, Singular solutions of a nonlinear equation in bounded domains of $\mathbb{R}^{2}$, J. Math. Anal. Appl., 270 (2002), 230-246.  doi: 10.1016/S0022-247X(02)00069-0.  Google Scholar

[20]

H. Mâagli and M. Zribi, On a semilinear fractional Dirichlet problem on a bounded domain, Appl. Math. Comput., 222 (2013), 331-342.  doi: 10.1016/j.amc.2013.07.041.  Google Scholar

[21]

V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.  Google Scholar

[22]

V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, Vol. 6 (Hindawi Publ. Corp., 2008). doi: 10.1155/9789774540394.  Google Scholar

[23]

D. Repovš, Singular solutions of perturbed logistic-type equations, Appl. Math. Comput., 218 (2011), 4414-4422.  doi: 10.1016/j.amc.2011.10.018.  Google Scholar

[24] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York-Berlin, 1987.  doi: 10.1007/978-0-387-75953-1.  Google Scholar
[25]

M. Selmi, Inequalities for Green functions in a Dini-Jordan domain in $\mathbb{R}^{2}$, Potential Anal., 13 (2000), 81-102.  doi: 10.1023/A:1008610631562.  Google Scholar

[26]

R. Seneta, Regularly Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag, Berlin, 1976.  Google Scholar

[27]

L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5 (1981), 225-242.  doi: 10.1016/0362-546X(81)90028-6.  Google Scholar

[28]

N. Zeddini, Positive solutions for a singular ponlinear Problem on a Bounded Domain in $\mathbb{R}^{2}$, Potential Anal., 18 (2003), 97-118.  doi: 10.1023/A:1020559619108.  Google Scholar

[29]

Q. S. Zhang and Z. Zhao, Singular solutions of semilinear elliptic and parabolic equations, Math. Ann., 310 (1998), 777-794.  doi: 10.1007/s002080050170.  Google Scholar

show all references

References:
[1]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl., vol. 27, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[2]

J. Bliedtner and W. Hansen, Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, 1986. doi: 10.1007/978-3-642-71131-2.  Google Scholar

[3]

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

[4]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, Third edition. Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.  Google Scholar

[5]

R. Chemmam, H. Mâagli, S. Masmoudi and M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal., 1 (2012), 301-318.  doi: 10.1515/anona-2012-0008.  Google Scholar

[6] K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer-Verlag, 1995.  doi: 10.1007/978-3-642-57856-4.  Google Scholar
[7]

F. Cirstea and V. D. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Math. Acad. Sci. Paris., 335 (2002), 447-452.  doi: 10.1016/S1631-073X(02)02503-7.  Google Scholar

[8]

F. Cirstea and V. D. Rădulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Transactions Amer. Math. Soc., 359 (2007), 3275-3286.  doi: 10.1090/S0002-9947-07-04107-4.  Google Scholar

[9]

M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar

[10]

S. Dumont, L. Dupaigne, O. Goubet and V. D. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.  doi: 10.1515/ans-2007-0205.  Google Scholar

[11]

M. Ghergu and V. D. Rădulescu, PDEs Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer Verlag, Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.  Google Scholar

[12]

M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Applications, Vol. 37, Oxford University Press, 2008.  Google Scholar

[13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, third ed., Springer Verlag, Berlin, 1983.  doi: 10.1007/978-3-642-61798-0.  Google Scholar
[14]

J. Karamata, Sur un mode de croissance régulière. Thé orèmes fondamentaux, Bull. Soc. Math. France., 61 (1933), 55-62.   Google Scholar

[15]

S. Karntz and S. Stević, On the iterated logarithmic Bloch space on the unit ball, Nonlinear Anal. TMA., 71 (2009), 1772-1795.  doi: 10.1016/j.na.2009.01.013.  Google Scholar

[16]

A. C. Lazer and P. J. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.  doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar

[17]

S. Li and S. Stević, On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces, Appl. Math. Comput., 215 (2009), 3106-3115.  doi: 10.1016/j.amc.2009.10.004.  Google Scholar

[18]

H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal., 74 (2011), 2941-2947.  doi: 10.1016/j.na.2011.01.011.  Google Scholar

[19]

H. Mâagli and L. Mâatoug, Singular solutions of a nonlinear equation in bounded domains of $\mathbb{R}^{2}$, J. Math. Anal. Appl., 270 (2002), 230-246.  doi: 10.1016/S0022-247X(02)00069-0.  Google Scholar

[20]

H. Mâagli and M. Zribi, On a semilinear fractional Dirichlet problem on a bounded domain, Appl. Math. Comput., 222 (2013), 331-342.  doi: 10.1016/j.amc.2013.07.041.  Google Scholar

[21]

V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.  Google Scholar

[22]

V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, Vol. 6 (Hindawi Publ. Corp., 2008). doi: 10.1155/9789774540394.  Google Scholar

[23]

D. Repovš, Singular solutions of perturbed logistic-type equations, Appl. Math. Comput., 218 (2011), 4414-4422.  doi: 10.1016/j.amc.2011.10.018.  Google Scholar

[24] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York-Berlin, 1987.  doi: 10.1007/978-0-387-75953-1.  Google Scholar
[25]

M. Selmi, Inequalities for Green functions in a Dini-Jordan domain in $\mathbb{R}^{2}$, Potential Anal., 13 (2000), 81-102.  doi: 10.1023/A:1008610631562.  Google Scholar

[26]

R. Seneta, Regularly Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag, Berlin, 1976.  Google Scholar

[27]

L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5 (1981), 225-242.  doi: 10.1016/0362-546X(81)90028-6.  Google Scholar

[28]

N. Zeddini, Positive solutions for a singular ponlinear Problem on a Bounded Domain in $\mathbb{R}^{2}$, Potential Anal., 18 (2003), 97-118.  doi: 10.1023/A:1020559619108.  Google Scholar

[29]

Q. S. Zhang and Z. Zhao, Singular solutions of semilinear elliptic and parabolic equations, Math. Ann., 310 (1998), 777-794.  doi: 10.1007/s002080050170.  Google Scholar

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