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 and 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, Electronic J. Diff. Equations, 90 (2009), 1-15.

[2]

C. Anedda and G. Porru, Second-order boundary estimates for solutions to singular elliptic equations in borderline cases, Electronic J. Diff. Equations, 51 (2009), 1-19.

[3]

S. Berhanu, F. Gladiali and G. Porru, Qualitative properties of solutions to elliptic singular problems, J. Inequal. Appl., 3 (1999), 313-330. doi: 10.1155/S1025583499000223.

[4]

S. Berhanu, F. Cuccu and G. Porru, On the boundary behaviour, including second order effects, of solutions to elliptic singular problems, Acta Mathematica Sinica (English Series), 23 (2007), 479-486. doi: 10.1007/s10114-005-0680-8.

[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, Nonlinear Anal., 71 (2009), 4137-4150. doi: 10.1016/j.na.2009.02.073.

[6]

N. Bingham, C. Goldie and J. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, 1987. doi: 10.1017/CBO9780511721434.

[7]

M. Crandall, P. Rabinowitz and L. Tartar, Dirichlet problem with a singular nonlinearity, Comm. Partial Diff. Equations, 2 (1977), 193-222.

[8]

F. Cuccu, E. Giarrusso and G. Porru, Boundary behaviour for solutions of elliptic singular equations with a gradient term, Nonlinear Anal., 69 (2008), 4550-4566. doi: 10.1016/j.na.2007.11.011.

[9]

F. Cîrstea and V. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, Sér. I, 335 (2002), 447-452. doi: 10.1016/S1631-073X(02)02503-7.

[10]

F. Cîrstea and V. Rădulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris, Sér. I, 336 (2003), 231-236. doi: 10.1016/S1631-073X(03)00027-X.

[11]

F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptot. Anal., 46 (2006), 275-298.

[12]

W. Fulks and J. Maybee, A singular nonlinear elliptic equation, Osaka J. Math., 12 (1960), 1-19.

[13]

E. Giarrusso and G. Porru, Boundary behaviour of solutions to nonlinear elliptic singular problems, Appl. Math. in the Golden Age, edited by J. C. Misra, Narosa Publishing House, New Dalhi, India, 2003, 163-178.

[14]

M. Ghergu and V. Rădulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation, C. R. Acad. Sci. Paris, Ser. I, 337 (2003), 259-264. doi: 10.1016/S1631-073X(03)00335-2.

[15]

C. Gui and F. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh, 123 A (1993), 1021-1029. doi: 10.1017/S030821050002970X.

[16]

E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term, Nonlinear Anal., 65 (2006), 107-128. doi: 10.1016/j.na.2005.08.007.

[17]

S. Gontara, H. Mâagli, S. Masmoudi and S. Turki, Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J. Math. Anal. Appl., 369 (2010), 719-729. doi: 10.1016/j.jmaa.2010.04.008.

[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, Adv. Nonlinear Studies, 7 (2007), 475-490.

[19]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3nd edition, Springer-Verlag, Berlin, 1998.

[20]

A. Lazer and P. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.2307/2048410.

[21]

A. Lair and A. Shaker, Classical and weak solutions of a singular elliptic problem, J. Math. Anal Appl., 211 (1997), 371-385. doi: 10.1006/jmaa.1997.5470.

[22]

P. McKenna and W. Reichel, Sign changing solutions to singular second order boundary value problem, Adv. in Differential Equations, 6 (2001), 441-460.

[23]

V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.

[24]

L. Mi and B. Liu, The second order estimate for the solution to a singular elliptic boundary value problem, Appl. Anal. Discrete Math., 6 (2012), 194-213. doi: 10.2298/AADM120713018M.

[25]

A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary value, J. Math. Anal. Appl., 325 (2007), 480-489. doi: 10.1016/j.jmaa.2006.02.008.

[26]

A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024.

[27]

G. Porru, A. Vitolo, Problems for elliptic singular equations with a quadratic gradient term, J. Math. Anal. Appl., 334 (2007), 467-486. doi: 10.1016/j.jmaa.2006.12.017.

[28]

S. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, Berlin, 1987. doi: 10.1007/978-0-387-75953-1.

[29]

C. Stuart, Existence and approximation of solutions of nonlinear elliptic equations, Math. Z., 147 (1976), 53-63.

[30]

J. Shi and M. Yao, On a singular semiinear elliptic problem, Proc. Roy. Soc. Edinburgh, 128 A (1998), 1389-1401. doi: 10.1017/S0308210500027384.

[31]

J. Shi and M. Yao, Positive solutions of elliptic equations with singular nonlinearity, Electronic J. Diff. Equations, 4 (2005), 1-11.

[32]

R. Seneta, Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976.

[33]

N. Zeddini, R. Alsaedi and H. Mâagli, Exact boundary behavior of the unique positive solution to some singular elliptic problems, Nonlinear Analysis, 89 (2013), 146-156. doi: 10.1016/j.na.2013.05.006.

[34]

Z. Zhang, The second expansion of the solution for a singular elliptic boundary value problems, J. Math. Anal. Appl., 381 (2011), 922-934. doi: 10.1016/j.jmaa.2011.04.018.

[35]

Z. Zhang and B. Li, The boundary behavior of the unique solution to a singular Dirichlet problem, J. Math. Anal. Appl., 391 (2012), 278-290. doi: 10.1016/j.jmaa.2012.02.010.

[36]

Z. Zhang, The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equations, J. Math. Anal. Appl., 312 (2005), 33-43. doi: 10.1016/j.jmaa.2005.03.023.

show all references

References:
[1]

C. Anedda, Second-order boundary estimates for solutions to singular elliptic equations, Electronic J. Diff. Equations, 90 (2009), 1-15.

[2]

C. Anedda and G. Porru, Second-order boundary estimates for solutions to singular elliptic equations in borderline cases, Electronic J. Diff. Equations, 51 (2009), 1-19.

[3]

S. Berhanu, F. Gladiali and G. Porru, Qualitative properties of solutions to elliptic singular problems, J. Inequal. Appl., 3 (1999), 313-330. doi: 10.1155/S1025583499000223.

[4]

S. Berhanu, F. Cuccu and G. Porru, On the boundary behaviour, including second order effects, of solutions to elliptic singular problems, Acta Mathematica Sinica (English Series), 23 (2007), 479-486. doi: 10.1007/s10114-005-0680-8.

[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, Nonlinear Anal., 71 (2009), 4137-4150. doi: 10.1016/j.na.2009.02.073.

[6]

N. Bingham, C. Goldie and J. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, 1987. doi: 10.1017/CBO9780511721434.

[7]

M. Crandall, P. Rabinowitz and L. Tartar, Dirichlet problem with a singular nonlinearity, Comm. Partial Diff. Equations, 2 (1977), 193-222.

[8]

F. Cuccu, E. Giarrusso and G. Porru, Boundary behaviour for solutions of elliptic singular equations with a gradient term, Nonlinear Anal., 69 (2008), 4550-4566. doi: 10.1016/j.na.2007.11.011.

[9]

F. Cîrstea and V. Rădulescu, Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C. R. Acad. Sci. Paris, Sér. I, 335 (2002), 447-452. doi: 10.1016/S1631-073X(02)02503-7.

[10]

F. Cîrstea and V. Rădulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris, Sér. I, 336 (2003), 231-236. doi: 10.1016/S1631-073X(03)00027-X.

[11]

F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptot. Anal., 46 (2006), 275-298.

[12]

W. Fulks and J. Maybee, A singular nonlinear elliptic equation, Osaka J. Math., 12 (1960), 1-19.

[13]

E. Giarrusso and G. Porru, Boundary behaviour of solutions to nonlinear elliptic singular problems, Appl. Math. in the Golden Age, edited by J. C. Misra, Narosa Publishing House, New Dalhi, India, 2003, 163-178.

[14]

M. Ghergu and V. Rădulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation, C. R. Acad. Sci. Paris, Ser. I, 337 (2003), 259-264. doi: 10.1016/S1631-073X(03)00335-2.

[15]

C. Gui and F. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh, 123 A (1993), 1021-1029. doi: 10.1017/S030821050002970X.

[16]

E. Giarrusso and G. Porru, Problems for elliptic singular equations with a gradient term, Nonlinear Anal., 65 (2006), 107-128. doi: 10.1016/j.na.2005.08.007.

[17]

S. Gontara, H. Mâagli, S. Masmoudi and S. Turki, Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J. Math. Anal. Appl., 369 (2010), 719-729. doi: 10.1016/j.jmaa.2010.04.008.

[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, Adv. Nonlinear Studies, 7 (2007), 475-490.

[19]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 3nd edition, Springer-Verlag, Berlin, 1998.

[20]

A. Lazer and P. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.2307/2048410.

[21]

A. Lair and A. Shaker, Classical and weak solutions of a singular elliptic problem, J. Math. Anal Appl., 211 (1997), 371-385. doi: 10.1006/jmaa.1997.5470.

[22]

P. McKenna and W. Reichel, Sign changing solutions to singular second order boundary value problem, Adv. in Differential Equations, 6 (2001), 441-460.

[23]

V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.

[24]

L. Mi and B. Liu, The second order estimate for the solution to a singular elliptic boundary value problem, Appl. Anal. Discrete Math., 6 (2012), 194-213. doi: 10.2298/AADM120713018M.

[25]

A. Mohammed, Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary value, J. Math. Anal. Appl., 325 (2007), 480-489. doi: 10.1016/j.jmaa.2006.02.008.

[26]

A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024.

[27]

G. Porru, A. Vitolo, Problems for elliptic singular equations with a quadratic gradient term, J. Math. Anal. Appl., 334 (2007), 467-486. doi: 10.1016/j.jmaa.2006.12.017.

[28]

S. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, Berlin, 1987. doi: 10.1007/978-0-387-75953-1.

[29]

C. Stuart, Existence and approximation of solutions of nonlinear elliptic equations, Math. Z., 147 (1976), 53-63.

[30]

J. Shi and M. Yao, On a singular semiinear elliptic problem, Proc. Roy. Soc. Edinburgh, 128 A (1998), 1389-1401. doi: 10.1017/S0308210500027384.

[31]

J. Shi and M. Yao, Positive solutions of elliptic equations with singular nonlinearity, Electronic J. Diff. Equations, 4 (2005), 1-11.

[32]

R. Seneta, Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976.

[33]

N. Zeddini, R. Alsaedi and H. Mâagli, Exact boundary behavior of the unique positive solution to some singular elliptic problems, Nonlinear Analysis, 89 (2013), 146-156. doi: 10.1016/j.na.2013.05.006.

[34]

Z. Zhang, The second expansion of the solution for a singular elliptic boundary value problems, J. Math. Anal. Appl., 381 (2011), 922-934. doi: 10.1016/j.jmaa.2011.04.018.

[35]

Z. Zhang and B. Li, The boundary behavior of the unique solution to a singular Dirichlet problem, J. Math. Anal. Appl., 391 (2012), 278-290. doi: 10.1016/j.jmaa.2012.02.010.

[36]

Z. Zhang, The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equations, J. Math. Anal. Appl., 312 (2005), 33-43. doi: 10.1016/j.jmaa.2005.03.023.

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