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: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
H. Brezis
and L. Oswald
, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986)
, 55-64.
doi: 10.1016/0362-546X(86)90011-8.![]() ![]() ![]() |
|
R. F. Brown,
A Topological Introduction to Nonlinear Analysis, Third edition. Springer, Cham, 2014.
doi: 10.1007/978-3-319-11794-2.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer-Verlag, 1995.
doi: 10.1007/978-3-642-57856-4.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. Karamata
, Sur un mode de croissance régulière. Thé orèmes fondamentaux, Bull. Soc. Math. France., 61 (1933)
, 55-62.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
V. Maric,
Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103952.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
D. Repovš
, Singular solutions of perturbed logistic-type equations, Appl. Math. Comput., 218 (2011)
, 4414-4422.
doi: 10.1016/j.amc.2011.10.018.![]() ![]() ![]() |
|
S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York-Berlin, 1987.
doi: 10.1007/978-0-387-75953-1.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
R. Seneta,
Regularly Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag, Berlin, 1976.
![]() ![]() |
|
L. Véron
, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5 (1981)
, 225-242.
doi: 10.1016/0362-546X(81)90028-6.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
Q. S. Zhang
and Z. Zhao
, Singular solutions of semilinear elliptic and parabolic equations, Math. Ann., 310 (1998)
, 777-794.
doi: 10.1007/s002080050170.![]() ![]() ![]() |