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

[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.  Google Scholar

[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.  Google Scholar

[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.  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, Nonlinear Anal., 71 (2009), 4137-4150. 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, Cambridge University Press, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. 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, Appl. Anal. Discrete Math., 6 (2012), 194-213. doi: 10.2298/AADM120713018M.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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, Nonlinear Anal., 71 (2009), 4137-4150. 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, Cambridge University Press, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

V. Maric, Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. 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, Appl. Anal. Discrete Math., 6 (2012), 194-213. doi: 10.2298/AADM120713018M.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[2]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[3]

Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197

[4]

Akisato Kubo. Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 59-74. doi: 10.3934/cpaa.2004.3.59

[5]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027

[6]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017

[7]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179

[8]

Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765

[9]

Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2067-2084. doi: 10.3934/dcdsb.2020325

[10]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212

[11]

Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251

[12]

Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431

[13]

Hideo Kubo. Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Conference Publications, 2007, 2007 (Special) : 602-613. doi: 10.3934/proc.2007.2007.602

[14]

Kazuhiro Ishige, Tatsuki Kawakami. Asymptotic behavior of solutions for some semilinear heat equations in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1351-1371. doi: 10.3934/cpaa.2009.8.1351

[15]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

[16]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619

[17]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701

[18]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247

[19]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

[20]

Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]