American Institute of Mathematical Sciences

Advanced
May  2013, 12(3): 1381-1392. doi: 10.3934/cpaa.2013.12.1381

Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior

Zhijun Zhang 1,

1. 

Department of Mathematics and Informational Science, Yantai University, P.O. Box 264005, Yantai, Shandong

Received  April 2012 Revised  July 2012 Published  September 2012

In this paper, we study the existence and boundary behavior of solutions to boundary blow-up elliptic problems \begin{eqnarray*} \triangle u =b(x)f(u)(1+|\nabla u|^q), u\geq 0, \ x\in \Omega,\ u|_{\partial \Omega}=\infty, \end{eqnarray*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $q\in (0, 2]$, $b \in C^{\alpha}(\bar{\Omega})$ which is positive in $\Omega$, may be vanishing on the boundary, and $f$ is normalised regularly varying at infinity with positive index $p$ and $p+q>1$.
Keywords: Semilinear elliptic equations, boundary behavior of solutions, boundary blow-up, a gradient term, uniqueness., existence.
Mathematics Subject Classification: Primary: 35J25, 35J65, 35J6.
Citation: 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
References:
[1]

S. Alarcón, J. García-Melián and A. Quaas, Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, 252 (2012), 886-914. doi: 10.1016/j.jde.2011.09.033.  Google Scholar

[2]

H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295. doi: 10.1007/BF01221152.  Google Scholar

[3]

C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math., 58 (1992), 9-24. doi: 10.1007/BF02790355.  Google Scholar

[4]

C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv. Differential Equations, 1 (1996), 133-150.  Google Scholar

[5]

E. B. Castillo and R. L. Albornoz, Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations, J. Math. Anal. Appl., 280 (2003), 123-132. doi: 10.1016/S0022-247X(03)00058-1.  Google Scholar

[6]

Y. Chen and M. Wang, Large solutions for quasilinear elliptic equation with nonlinear gradient term, Nonlinear Anal.: Real World Appl., 12 (2011), 455-463. doi: 10.1016/j.nonrwa.2010.06.031.  Google Scholar

[7]

Y. Chen and M. Wang, Boundary blow-up solutions for elliptic equations with gradient terms and singular weights : existence, asymptotic behaviour and uniqueness, Proc. Roy. Soc. Edinb., 141A (2011), 717-737.  Google Scholar

[8]

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/FLA.  Google Scholar

[9]

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

[10]

F. Cîrstea and Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc. London Math. Soc., 91 (2005), 459-482. doi: 10.1112/S0024611505015273.  Google Scholar

[11]

V. Ferone, Boundary blow-up for nonlinear elliptic equations with general growth in the gradient: an approach via symmetrisation, Le Matematiche, 65 (2010), 55-68. doi: 10.4418/2010.65.2.8.  Google Scholar

[12]

V. Ferone, E. Giarrusso, B. Messano and M. R. Posteraro, Estimates for blow-up solutions to nonlinear elliptic equations with $p$-growth in the gradient, Z. Anal. Anwend., 29 (2010), 219-234. doi: 10.4171/ZAA/1406.  Google Scholar

[13]

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

[14]

E. Giarrusso, Asymptotic behavior of large solutions of an elliptic quasilinear equation in a borderline case, C.R. Acad. Sci. Paris Ser. I, 331 (2000), 777-782. doi: 10.1016/S0764-4442(00)01707-9/FLA.  Google Scholar

[15]

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

[16]

V. Goncalves, A. Roncalli, Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput., 182 (2006), 13-23. doi: 10.1016/j.amc.2006.03.029.  Google Scholar

[17]

S. Huang, W. Li, Q. Tian and C. Mu, Large solution to nonlinear elliptic equation with nonlinear gradient terms, J. Diff. Equations, 251 (2011), 3297-3328. doi: 10.1016/j.jde.2011.08.031.  Google Scholar

[18]

J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar

[19]

A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic equations with nonlinear gradient terms, Int. J. Math. Math. Sci., 22 (1999), 869-883. doi: 10.1006/S 0161-1712<99>22869-4.  Google Scholar

[20]

A. V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl., 240 (1999), 205-218. doi: 10.1006/jmaa.1999.6609.  Google Scholar

[21]

J. M. Lasry and P. L. Lions, Nonlinear elliptic equations with singular boundary Conditions and stochastic control with state constrains, Math. Ann., 283 (1989), 583-630. doi: 10.1007/BF01442856.  Google Scholar

[22]

T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Studies, 7 (2007), 237-269.  Google Scholar

[23]

T. Leonori and A. Porretta, The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint, SIAM J. Math. Anal., 39 (2007), 1295-1327. doi: 10.1137/070681363.  Google Scholar

[24]

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

[25]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar

[26]

A. Porretta and L. Véron, Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations, Adv. Nonlinear Studies, 6 (2006), 351-378.  Google Scholar

[27]

S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'' Springer-Verlag, New York, Berlin, 1987.  Google Scholar

[28]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms, J. Differential Equations, 228 (2006), 661-684. doi: 10.1016/j.jde.2006.02.003.  Google Scholar

[29]

Z. Zhang, Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Nonlinear Anal., 73 (2010), 3348-3363. doi: 10.1016/j.na.2010.07.017.  Google Scholar

[30]

Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. doi: 10.1006//jmaa.1997.5635.  Google Scholar

[31]

Z. Zhang, Y. Ma, L. Mi and X. Li, Blow-up rates of large solutions for elliptic equations, J. Differential Equations, 249 (2010), 180-199. doi: 10.1016/j.jde.2010.02.019.  Google Scholar

show all references

References:
[1]

S. Alarcón, J. García-Melián and A. Quaas, Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, 252 (2012), 886-914. doi: 10.1016/j.jde.2011.09.033.  Google Scholar

[2]

H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295. doi: 10.1007/BF01221152.  Google Scholar

[3]

C. Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Analyse Math., 58 (1992), 9-24. doi: 10.1007/BF02790355.  Google Scholar

[4]

C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv. Differential Equations, 1 (1996), 133-150.  Google Scholar

[5]

E. B. Castillo and R. L. Albornoz, Local gradient estimates and existence of blow-up solutions to a class of quasilinear elliptic equations, J. Math. Anal. Appl., 280 (2003), 123-132. doi: 10.1016/S0022-247X(03)00058-1.  Google Scholar

[6]

Y. Chen and M. Wang, Large solutions for quasilinear elliptic equation with nonlinear gradient term, Nonlinear Anal.: Real World Appl., 12 (2011), 455-463. doi: 10.1016/j.nonrwa.2010.06.031.  Google Scholar

[7]

Y. Chen and M. Wang, Boundary blow-up solutions for elliptic equations with gradient terms and singular weights : existence, asymptotic behaviour and uniqueness, Proc. Roy. Soc. Edinb., 141A (2011), 717-737.  Google Scholar

[8]

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/FLA.  Google Scholar

[9]

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

[10]

F. Cîrstea and Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc. London Math. Soc., 91 (2005), 459-482. doi: 10.1112/S0024611505015273.  Google Scholar

[11]

V. Ferone, Boundary blow-up for nonlinear elliptic equations with general growth in the gradient: an approach via symmetrisation, Le Matematiche, 65 (2010), 55-68. doi: 10.4418/2010.65.2.8.  Google Scholar

[12]

V. Ferone, E. Giarrusso, B. Messano and M. R. Posteraro, Estimates for blow-up solutions to nonlinear elliptic equations with $p$-growth in the gradient, Z. Anal. Anwend., 29 (2010), 219-234. doi: 10.4171/ZAA/1406.  Google Scholar

[13]

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

[14]

E. Giarrusso, Asymptotic behavior of large solutions of an elliptic quasilinear equation in a borderline case, C.R. Acad. Sci. Paris Ser. I, 331 (2000), 777-782. doi: 10.1016/S0764-4442(00)01707-9/FLA.  Google Scholar

[15]

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

[16]

V. Goncalves, A. Roncalli, Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput., 182 (2006), 13-23. doi: 10.1016/j.amc.2006.03.029.  Google Scholar

[17]

S. Huang, W. Li, Q. Tian and C. Mu, Large solution to nonlinear elliptic equation with nonlinear gradient terms, J. Diff. Equations, 251 (2011), 3297-3328. doi: 10.1016/j.jde.2011.08.031.  Google Scholar

[18]

J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar

[19]

A. V. Lair and A. W. Wood, Large solutions of semilinear elliptic equations with nonlinear gradient terms, Int. J. Math. Math. Sci., 22 (1999), 869-883. doi: 10.1006/S 0161-1712<99>22869-4.  Google Scholar

[20]

A. V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl., 240 (1999), 205-218. doi: 10.1006/jmaa.1999.6609.  Google Scholar

[21]

J. M. Lasry and P. L. Lions, Nonlinear elliptic equations with singular boundary Conditions and stochastic control with state constrains, Math. Ann., 283 (1989), 583-630. doi: 10.1007/BF01442856.  Google Scholar

[22]

T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Studies, 7 (2007), 237-269.  Google Scholar

[23]

T. Leonori and A. Porretta, The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint, SIAM J. Math. Anal., 39 (2007), 1295-1327. doi: 10.1137/070681363.  Google Scholar

[24]

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

[25]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar

[26]

A. Porretta and L. Véron, Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations, Adv. Nonlinear Studies, 6 (2006), 351-378.  Google Scholar

[27]

S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'' Springer-Verlag, New York, Berlin, 1987.  Google Scholar

[28]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms, J. Differential Equations, 228 (2006), 661-684. doi: 10.1016/j.jde.2006.02.003.  Google Scholar

[29]

Z. Zhang, Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Nonlinear Anal., 73 (2010), 3348-3363. doi: 10.1016/j.na.2010.07.017.  Google Scholar

[30]

Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. doi: 10.1006//jmaa.1997.5635.  Google Scholar

[31]

Z. Zhang, Y. Ma, L. Mi and X. Li, Blow-up rates of large solutions for elliptic equations, J. Differential Equations, 249 (2010), 180-199. doi: 10.1016/j.jde.2010.02.019.  Google Scholar

[1]

Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[2]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[3]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[4]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[5]

Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549
[6]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828
[7]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[8]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[9]

Mingzhu Wu, Zuodong Yang. Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case. Communications on Pure & Applied Analysis, 2007, 6 (2) : 531-540. doi: 10.3934/cpaa.2007.6.531
[10]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[11]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[12]

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435
[13]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[14]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[15]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039
[16]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[17]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023
[18]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271
[19]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[20]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences