May  2015, 14(3): 1053-1072. doi: 10.3934/cpaa.2015.14.1053

Asymptotic behavior for the unique positive solution to a singular elliptic problem

1. 

School of Science, Linyi University, Linyi, Shandong, 276005, China

Received  October 2014 Revised  December 2014 Published  March 2015

In this paper, by means of sub-supersolution method, we are concerned with the exact asymptotic behavior for the unique solution near the boundary to the following singular Dirichlet problem $ -\triangle u=b(x)g(u), u>0, x \in \Omega, u|_{\partial \Omega}=0$, where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $b \in C^{\alpha}_{l o c}({\Omega})$ ($0 < \alpha < 1$), is positive in $\Omega,$ may be vanishing or singular on the boundary, $g\in C^1((0,\infty), (0,\infty))$, $g$ is decreasing on $(0,\infty)$ with $\lim\limits_{s \rightarrow 0^+}g(s)=\infty$ and satisfies some appropriate assumptions related to Karamata regular variation theory.
Citation: Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053
References:
[1]

C. Anedda, Second-order boundary estimates for solutions to singular elliptic equations,, \emph{Electronic J. Diff. Equations}, 90 (2009), 1.   Google Scholar

[2]

C. Anedda and G. Porru, Second-order boundary estimates for solutions to singular elliptic equations in borderline cases,, \emph{Electronic J. Diff. Equations}, 51 (2009), 1.   Google Scholar

[3]

S. Berhanu, F. Gladiali and G. Porru, Qualitative properties of solutions to elliptic singular problems,, \emph{J. Inequal. Appl.}, 3 (1999), 313.  doi: 10.1155/S1025583499000223.  Google Scholar

[4]

S. Berhanu, F. Cuccu and G. Porru, On the boundary behaviour, including second order effects, of solutions to elliptic singular problems,, \emph{Acta Mathematica Sinica (English Series)}, 23 (2007), 479.  doi: 10.1007/s10114-005-0680-8.  Google Scholar

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S. Ben Othman, H. Mâagli, S. Masmoudi and M. Zribi, Exact asymptotic behaviour near the boundary to the solution for singular nonlinear Dirichlet problems,, \emph{Nonlinear Anal.}, 71 (2009), 4137.  doi: 10.1016/j.na.2009.02.073.  Google Scholar

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N. Bingham, C. Goldie and J. Teugels, Regular Variation,, Encyclopedia of Mathematics and its Applications 27, (1987).  doi: 10.1017/CBO9780511721434.  Google Scholar

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M. Crandall, P. Rabinowitz and L. Tartar, Dirichlet problem with a singular nonlinearity,, \emph{Comm. Partial Diff. Equations}, 2 (1977), 193.   Google Scholar

[8]

F. Cuccu, E. Giarrusso and G. Porru, Boundary behaviour for solutions of elliptic singular equations with a gradient term,, \emph{Nonlinear Anal.}, 69 (2008), 4550.  doi: 10.1016/j.na.2007.11.011.  Google Scholar

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F. Cîrstea and V. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion,, \emph{C. R. Acad. Sci. Paris, 335 (2002), 447.  doi: 10.1016/S1631-073X(02)02503-7.  Google Scholar

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F. Cîrstea and V. Rădulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption,, \emph{C. R. Acad. Sci. Paris, 336 (2003), 231.  doi: 10.1016/S1631-073X(03)00027-X.  Google Scholar

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F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,, \emph{Asymptot. Anal.}, 46 (2006), 275.   Google Scholar

[12]

W. Fulks and J. Maybee, A singular nonlinear elliptic equation,, \emph{Osaka J. Math.}, 12 (1960), 1.   Google Scholar

[13]

E. Giarrusso and G. Porru, Boundary behaviour of solutions to nonlinear elliptic singular problems,, \emph{Appl. Math. in the Golden Age}, (2003), 163.   Google Scholar

[14]

M. Ghergu and V. Rădulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation,, \emph{C. R. Acad. Sci. Paris, 337 (2003), 259.  doi: 10.1016/S1631-073X(03)00335-2.  Google Scholar

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C. Gui and F. Lin, Regularity of an elliptic problem with a singular nonlinearity,, \emph{Proc. Roy. Soc. Edinburgh}, 123 A (1993), 1021.  doi: 10.1017/S030821050002970X.  Google Scholar

[16]

E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term,, \emph{ Nonlinear Anal.}, 65 (2006), 107.  doi: 10.1016/j.na.2005.08.007.  Google Scholar

[17]

S. Gontara, H. Mâagli, S. Masmoudi and S. Turki, Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem,, \emph{J. Math. Anal. Appl.}, 369 (2010), 719.  doi: 10.1016/j.jmaa.2010.04.008.  Google Scholar

[18]

J. Goncalves, A. Melo and C. Santos, On existence of $L^\infty$-ground states for singular elliptic equations in the presence of a strongly nonlinear term,, \emph{Adv. Nonlinear Studies}, 7 (2007), 475.   Google Scholar

[19]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, 3nd edition, (1998).   Google Scholar

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A. Lazer and P. McKenna, On a singular elliptic boundary value problem,, \emph{Proc. Amer. Math. Soc.}, 111 (1991), 721.  doi: 10.2307/2048410.  Google Scholar

[21]

A. Lair and A. Shaker, Classical and weak solutions of a singular elliptic problem,, \emph{J. Math. Anal Appl.}, 211 (1997), 371.  doi: 10.1006/jmaa.1997.5470.  Google Scholar

[22]

P. McKenna and W. Reichel, Sign changing solutions to singular second order boundary value problem,, \emph{Adv. in Differential Equations}, 6 (2001), 441.   Google Scholar

[23]

V. Maric, Regular Variation and Differential Equations,, Lecture Notes in Math., (1726).  doi: 10.1007/BFb0103952.  Google Scholar

[24]

L. Mi and B. Liu, The second order estimate for the solution to a singular elliptic boundary value problem,, \emph{Appl. Anal. Discrete Math.}, 6 (2012), 194.  doi: 10.2298/AADM120713018M.  Google Scholar

[25]

A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary value,, \emph{J. Math. Anal. Appl.}, 325 (2007), 480.  doi: 10.1016/j.jmaa.2006.02.008.  Google Scholar

[26]

A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids,, \emph{SIAM J. Appl. Math.}, 38 (1980), 275.  doi: 10.1137/0138024.  Google Scholar

[27]

G. Porru, A. Vitolo, Problems for elliptic singular equations with a quadratic gradient term,, \emph{J. Math. Anal. Appl.}, 334 (2007), 467.  doi: 10.1016/j.jmaa.2006.12.017.  Google Scholar

[28]

S. Resnick, Extreme Values, Regular Variation, and Point Processes,, Springer-Verlag, (1987).  doi: 10.1007/978-0-387-75953-1.  Google Scholar

[29]

C. Stuart, Existence and approximation of solutions of nonlinear elliptic equations,, \emph{Math. Z.}, 147 (1976), 53.   Google Scholar

[30]

J. Shi and M. Yao, On a singular semiinear elliptic problem,, \emph{Proc. Roy. Soc. Edinburgh}, 128 A (1998), 1389.  doi: 10.1017/S0308210500027384.  Google Scholar

[31]

J. Shi and M. Yao, Positive solutions of elliptic equations with singular nonlinearity,, \emph{Electronic J. Diff. Equations}, 4 (2005), 1.   Google Scholar

[32]

R. Seneta, Regular Varying Functions,, Lecture Notes in Math., (1976).   Google Scholar

[33]

N. Zeddini, R. Alsaedi and H. Mâagli, Exact boundary behavior of the unique positive solution to some singular elliptic problems,, \emph{Nonlinear Analysis}, 89 (2013), 146.  doi: 10.1016/j.na.2013.05.006.  Google Scholar

[34]

Z. Zhang, The second expansion of the solution for a singular elliptic boundary value problems,, \emph{J. Math. Anal. Appl.}, 381 (2011), 922.  doi: 10.1016/j.jmaa.2011.04.018.  Google Scholar

[35]

Z. Zhang and B. Li, The boundary behavior of the unique solution to a singular Dirichlet problem,, \emph{J. Math. Anal. Appl.}, 391 (2012), 278.  doi: 10.1016/j.jmaa.2012.02.010.  Google Scholar

[36]

Z. Zhang, The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equations,, \emph{J. Math. Anal. Appl.}, 312 (2005), 33.  doi: 10.1016/j.jmaa.2005.03.023.  Google Scholar

show all references

References:
[1]

C. Anedda, Second-order boundary estimates for solutions to singular elliptic equations,, \emph{Electronic J. Diff. Equations}, 90 (2009), 1.   Google Scholar

[2]

C. Anedda and G. Porru, Second-order boundary estimates for solutions to singular elliptic equations in borderline cases,, \emph{Electronic J. Diff. Equations}, 51 (2009), 1.   Google Scholar

[3]

S. Berhanu, F. Gladiali and G. Porru, Qualitative properties of solutions to elliptic singular problems,, \emph{J. Inequal. Appl.}, 3 (1999), 313.  doi: 10.1155/S1025583499000223.  Google Scholar

[4]

S. Berhanu, F. Cuccu and G. Porru, On the boundary behaviour, including second order effects, of solutions to elliptic singular problems,, \emph{Acta Mathematica Sinica (English Series)}, 23 (2007), 479.  doi: 10.1007/s10114-005-0680-8.  Google Scholar

[5]

S. Ben Othman, H. Mâagli, S. Masmoudi and M. Zribi, Exact asymptotic behaviour near the boundary to the solution for singular nonlinear Dirichlet problems,, \emph{Nonlinear Anal.}, 71 (2009), 4137.  doi: 10.1016/j.na.2009.02.073.  Google Scholar

[6]

N. Bingham, C. Goldie and J. Teugels, Regular Variation,, Encyclopedia of Mathematics and its Applications 27, (1987).  doi: 10.1017/CBO9780511721434.  Google Scholar

[7]

M. Crandall, P. Rabinowitz and L. Tartar, Dirichlet problem with a singular nonlinearity,, \emph{Comm. Partial Diff. Equations}, 2 (1977), 193.   Google Scholar

[8]

F. Cuccu, E. Giarrusso and G. Porru, Boundary behaviour for solutions of elliptic singular equations with a gradient term,, \emph{Nonlinear Anal.}, 69 (2008), 4550.  doi: 10.1016/j.na.2007.11.011.  Google Scholar

[9]

F. Cîrstea and V. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion,, \emph{C. R. Acad. Sci. Paris, 335 (2002), 447.  doi: 10.1016/S1631-073X(02)02503-7.  Google Scholar

[10]

F. Cîrstea and V. Rădulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption,, \emph{C. R. Acad. Sci. Paris, 336 (2003), 231.  doi: 10.1016/S1631-073X(03)00027-X.  Google Scholar

[11]

F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,, \emph{Asymptot. Anal.}, 46 (2006), 275.   Google Scholar

[12]

W. Fulks and J. Maybee, A singular nonlinear elliptic equation,, \emph{Osaka J. Math.}, 12 (1960), 1.   Google Scholar

[13]

E. Giarrusso and G. Porru, Boundary behaviour of solutions to nonlinear elliptic singular problems,, \emph{Appl. Math. in the Golden Age}, (2003), 163.   Google Scholar

[14]

M. Ghergu and V. Rădulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation,, \emph{C. R. Acad. Sci. Paris, 337 (2003), 259.  doi: 10.1016/S1631-073X(03)00335-2.  Google Scholar

[15]

C. Gui and F. Lin, Regularity of an elliptic problem with a singular nonlinearity,, \emph{Proc. Roy. Soc. Edinburgh}, 123 A (1993), 1021.  doi: 10.1017/S030821050002970X.  Google Scholar

[16]

E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term,, \emph{ Nonlinear Anal.}, 65 (2006), 107.  doi: 10.1016/j.na.2005.08.007.  Google Scholar

[17]

S. Gontara, H. Mâagli, S. Masmoudi and S. Turki, Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem,, \emph{J. Math. Anal. Appl.}, 369 (2010), 719.  doi: 10.1016/j.jmaa.2010.04.008.  Google Scholar

[18]

J. Goncalves, A. Melo and C. Santos, On existence of $L^\infty$-ground states for singular elliptic equations in the presence of a strongly nonlinear term,, \emph{Adv. Nonlinear Studies}, 7 (2007), 475.   Google Scholar

[19]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, 3nd edition, (1998).   Google Scholar

[20]

A. Lazer and P. McKenna, On a singular elliptic boundary value problem,, \emph{Proc. Amer. Math. Soc.}, 111 (1991), 721.  doi: 10.2307/2048410.  Google Scholar

[21]

A. Lair and A. Shaker, Classical and weak solutions of a singular elliptic problem,, \emph{J. Math. Anal Appl.}, 211 (1997), 371.  doi: 10.1006/jmaa.1997.5470.  Google Scholar

[22]

P. McKenna and W. Reichel, Sign changing solutions to singular second order boundary value problem,, \emph{Adv. in Differential Equations}, 6 (2001), 441.   Google Scholar

[23]

V. Maric, Regular Variation and Differential Equations,, Lecture Notes in Math., (1726).  doi: 10.1007/BFb0103952.  Google Scholar

[24]

L. Mi and B. Liu, The second order estimate for the solution to a singular elliptic boundary value problem,, \emph{Appl. Anal. Discrete Math.}, 6 (2012), 194.  doi: 10.2298/AADM120713018M.  Google Scholar

[25]

A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary value,, \emph{J. Math. Anal. Appl.}, 325 (2007), 480.  doi: 10.1016/j.jmaa.2006.02.008.  Google Scholar

[26]

A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids,, \emph{SIAM J. Appl. Math.}, 38 (1980), 275.  doi: 10.1137/0138024.  Google Scholar

[27]

G. Porru, A. Vitolo, Problems for elliptic singular equations with a quadratic gradient term,, \emph{J. Math. Anal. Appl.}, 334 (2007), 467.  doi: 10.1016/j.jmaa.2006.12.017.  Google Scholar

[28]

S. Resnick, Extreme Values, Regular Variation, and Point Processes,, Springer-Verlag, (1987).  doi: 10.1007/978-0-387-75953-1.  Google Scholar

[29]

C. Stuart, Existence and approximation of solutions of nonlinear elliptic equations,, \emph{Math. Z.}, 147 (1976), 53.   Google Scholar

[30]

J. Shi and M. Yao, On a singular semiinear elliptic problem,, \emph{Proc. Roy. Soc. Edinburgh}, 128 A (1998), 1389.  doi: 10.1017/S0308210500027384.  Google Scholar

[31]

J. Shi and M. Yao, Positive solutions of elliptic equations with singular nonlinearity,, \emph{Electronic J. Diff. Equations}, 4 (2005), 1.   Google Scholar

[32]

R. Seneta, Regular Varying Functions,, Lecture Notes in Math., (1976).   Google Scholar

[33]

N. Zeddini, R. Alsaedi and H. Mâagli, Exact boundary behavior of the unique positive solution to some singular elliptic problems,, \emph{Nonlinear Analysis}, 89 (2013), 146.  doi: 10.1016/j.na.2013.05.006.  Google Scholar

[34]

Z. Zhang, The second expansion of the solution for a singular elliptic boundary value problems,, \emph{J. Math. Anal. Appl.}, 381 (2011), 922.  doi: 10.1016/j.jmaa.2011.04.018.  Google Scholar

[35]

Z. Zhang and B. Li, The boundary behavior of the unique solution to a singular Dirichlet problem,, \emph{J. Math. Anal. Appl.}, 391 (2012), 278.  doi: 10.1016/j.jmaa.2012.02.010.  Google Scholar

[36]

Z. Zhang, The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equations,, \emph{J. Math. Anal. Appl.}, 312 (2005), 33.  doi: 10.1016/j.jmaa.2005.03.023.  Google Scholar

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