American Institute of Mathematical Sciences

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

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

 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
 $0$
.
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

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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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