×

Singular solutions of a nonlinear equation in a punctured domain of \(\mathbb{R}^{2}\). (English) Zbl 1414.35090

Summary: We consider the following singular semilinear problem \[\begin{cases}-\Delta u(x)=a(x)u^\sigma(x),\quad x\in \Omega\setminus\{0\}\quad\text{(in the distributional sense)},\\ u>0, \text{ on }\Omega\setminus\{0\},\\ \lim_{\vert x\vert\to 0}\frac{u(x) }{\ln\vert x\vert}=0,\\ u(x)=0,\;x\in\partial \Omega, \end{cases}\] where \(\sigma <1\), \(\Omega\) is a bounded regular domain in \(\mathbb R^2\) with \(0\in\Omega\). The weight function \(a(x)\) is required to be positive and continuous in \(\Omega\setminus\{0\}\) with the possibility to be singular at \(x=0\) and/or at the boundary \(\partial \Omega\). When the function \(a\) satisfies sharp estimates related to Karamata class, we prove the existence and global asymptotic behavior of a positive continuous solution on \(\overline \Omega\) which could blow-up at \(0\).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J75 Singular elliptic equations
35B09 Positive solutions to PDEs
35B44 Blow-up in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl., vol. 27, Cambridge University Press, Cambridge, 1987. · Zbl 0617.26001
[2] J. Bliedtner and W. Hansen, Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, 1986. · Zbl 0706.31001
[3] H. Brezis; L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10, 55-64 (1986) · Zbl 0593.35045 · doi:10.1016/0362-546X(86)90011-8
[4] R. F. Brown, A Topological Introduction to Nonlinear Analysis, Third edition. Springer, Cham, 2014. · Zbl 1321.47001
[5] R. Chemmam; H. Mâagli; S. Masmoudi; M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal., 1, 301-318 (2012) · Zbl 1277.31016 · doi:10.1515/anona-2012-0008
[6] F. Cirstea; V. D. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Math. Acad. Sci. Paris., 335, 447-452 (2002) · Zbl 1183.35124 · doi:10.1016/S1631-073X(02)02503-7
[7] F. Cirstea; V. D. Rădulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Transactions Amer. Math. Soc., 359, 3275-3286 (2007) · Zbl 1134.35039 · doi:10.1090/S0002-9947-07-04107-4
[8] M. G. Crandall; P. H. Rabinowitz; L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., 2, 193-222 (1977) · Zbl 0362.35031 · doi:10.1080/03605307708820029
[9] S. Dumont; L. Dupaigne; O. Goubet; V. D. Rădulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7, 271-298 (2007) · Zbl 1137.35030 · doi:10.1515/ans-2007-0205
[10] M. Ghergu and V. D. Rădulescu, PDEs Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics, Springer Verlag, Heidelberg, 2012. · Zbl 1227.35001
[11] 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. · Zbl 1159.35030
[12] J. Karamata, Sur un mode de croissance régulière. Thé orèmes fondamentaux, Bull. Soc. Math. France., 61, 55-62 (1933) · JFM 59.0994.01
[13] S. Karntz; S. Stević, On the iterated logarithmic Bloch space on the unit ball, Nonlinear Anal. TMA., 71, 1772-1795 (2009) · Zbl 1221.47056 · doi:10.1016/j.na.2009.01.013
[14] A. C. Lazer; P. J. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111, 721-730 (1991) · Zbl 0727.35057 · doi:10.1090/S0002-9939-1991-1037213-9
[15] S. Li; S. Stević, On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces, Appl. Math. Comput., 215, 3106-3115 (2009) · Zbl 1181.30032 · doi:10.1016/j.amc.2009.10.004
[16] H. Mâagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal., 74, 2941-2947 (2011) · Zbl 1213.31006 · doi:10.1016/j.na.2011.01.011
[17] H. Mâagli; L. Mâatoug, Singular solutions of a nonlinear equation in bounded domains of \(\begin{document}\mathbb{R}^2\end{document} \), J. Math. Anal. Appl., 270, 230-246 (2002) · Zbl 1018.35036 · doi:10.1016/S0022-247X(02)00069-0
[18] H. Mâagli; M. Zribi, On a semilinear fractional Dirichlet problem on a bounded domain, Appl. Math. Comput., 222, 331-342 (2013) · Zbl 1330.35505 · doi:10.1016/j.amc.2013.07.041
[19] V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin, 2000. · Zbl 0946.34001
[20] 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). · Zbl 1190.35003
[21] D. Repovš, Singular solutions of perturbed logistic-type equations, Appl. Math. Comput., 218, 4414-4422 (2011) · Zbl 1239.35161 · doi:10.1016/j.amc.2011.10.018
[22] M. Selmi, Inequalities for Green functions in a Dini-Jordan domain in \(\begin{document}\mathbb{R}^2\end{document} \), Potential Anal., 13, 81-102 (2000) · Zbl 0959.31004 · doi:10.1023/A:1008610631562
[23] R. Seneta, Regularly Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag, Berlin, 1976. · Zbl 0324.26002
[24] L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5, 225-242 (1981) · Zbl 0457.35031 · doi:10.1016/0362-546X(81)90028-6
[25] N. Zeddini, Positive solutions for a singular ponlinear Problem on a Bounded Domain in \(\begin{document}\mathbb{R}^2\end{document} \), Potential Anal., 18, 97-118 (2003) · Zbl 1028.34030 · doi:10.1023/A:1020559619108
[26] Q. S. Zhang; Z. Zhao, Singular solutions of semilinear elliptic and parabolic equations, Math. Ann., 310, 777-794 (1998) · Zbl 0907.35047 · doi:10.1007/s002080050170
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.