Article Contents
Article Contents

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

• 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 $p+q>1$.
Mathematics Subject Classification: Primary: 35J25, 35J65, 35J67.

 Citation:

•  [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.