American Institute of Mathematical Sciences

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

 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 $p+q>1$.
Citation: Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure and 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. [2] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295. doi: 10.1007/BF01221152. [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. [4] C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv. Differential Equations, 1 (1996), 133-150. [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. [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. [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. [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. [9] F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptotic Anal., 46 (2006), 275-298. [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. [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. [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. [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. [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. [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 3nd edition, Springer - Verlag, Berlin, 1998. [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. [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. [18] J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. [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. [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. [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. [22] T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Studies, 7 (2007), 237-269. [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. [24] V. Maric, "Regular Variation and Differential Equations,'' Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952. [25] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. [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. [27] S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'' Springer-Verlag, New York, Berlin, 1987. [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. [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. [30] Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. doi: 10.1006//jmaa.1997.5635. [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.

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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. [2] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295. doi: 10.1007/BF01221152. [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. [4] C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equations with nonlinear gradient term, Adv. Differential Equations, 1 (1996), 133-150. [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. [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. [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. [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. [9] F. Cîrstea and V. Rădulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptotic Anal., 46 (2006), 275-298. [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. [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. [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. [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. [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. [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 3nd edition, Springer - Verlag, Berlin, 1998. [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. [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. [18] J. B. Keller, On solutions of $\Delta u=f(u)$, Commun. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402. [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. [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. [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. [22] T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. Nonlinear Studies, 7 (2007), 237-269. [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. [24] V. Maric, "Regular Variation and Differential Equations,'' Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952. [25] R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. [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. [27] S. I. Resnick, "Extreme Values, Regular Variation, and Point Processes,'' Springer-Verlag, New York, Berlin, 1987. [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. [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. [30] Z. Zhang, Nonlinear elliptic equations with singular boundary conditions, J. Math. Anal. Appl., 216 (1997), 390-397. doi: 10.1006//jmaa.1997.5635. [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.
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