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Nonlinear equations involving the square root of the Laplacian
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 |
$\left\{ \begin{array}{l} - \Delta u(x) = a(x){u^\sigma }(x),{\rm{ }}x \in \Omega \backslash \{ 0\} ({\rm{in\;the\;distributional\;sense}}),\\\;u > 0,{\rm{ on}}\;\Omega \backslash \{ 0\} ,\\\mathop {\lim }\limits_{\left| x \right| \to 0} \frac{{u(x)}}{{\ln \left| x \right|}} = 0,\\u(x) = 0,\;x \in \partial \Omega ,\end{array} \right.$ |
$σ <1,$ |
$Ω $ |
$\mathbb{R}^{2}$ |
$0∈ Ω .$ |
$a(x)$ |
$Ω \backslash \{0\}$ |
$x = 0$ |
$\partial Ω. $ |
$a$ |
$\overline{Ω }\backslash \{0\}$ |
$0$ |
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. |
[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. |
[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. |
[4] |
R. F. Brown,
A Topological Introduction to Nonlinear Analysis, Third edition. Springer, Cham, 2014.
doi: 10.1007/978-3-319-11794-2. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[14] |
J. Karamata,
Sur un mode de croissance régulière. Thé orèmes fondamentaux, Bull. Soc. Math. France., 61 (1933), 55-62.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
V. Maric,
Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103952. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[26] |
R. Seneta,
Regularly Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag, Berlin, 1976. |
[27] |
L. Véron,
Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5 (1981), 225-242.
doi: 10.1016/0362-546X(81)90028-6. |
[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. |
[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. |
show all references
Dedicated to Vicenţiu D. Rǎdulescu on his sixtieth birthday
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. |
[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. |
[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. |
[4] |
R. F. Brown,
A Topological Introduction to Nonlinear Analysis, Third edition. Springer, Cham, 2014.
doi: 10.1007/978-3-319-11794-2. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[14] |
J. Karamata,
Sur un mode de croissance régulière. Thé orèmes fondamentaux, Bull. Soc. Math. France., 61 (1933), 55-62.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
V. Maric,
Regular Variation and Differential Equations, Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103952. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[26] |
R. Seneta,
Regularly Varying Functions, Lectures Notes in Math., Vol. 508, Springer-Verlag, Berlin, 1976. |
[27] |
L. Véron,
Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5 (1981), 225-242.
doi: 10.1016/0362-546X(81)90028-6. |
[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. |
[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. |
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