Asymptotic behaviour of the gradient of large solutions to some nonlinear elliptic equations. (English) Zbl 1221.35141

Summary: If \(h\) is a nondecreasing real valued function and \(0\leq q\leq 2\), we analyse the boundary behaviour of the gradient of any solution \(u\) of \(-\Delta u+h(u)+|\nabla u|^q=f\) in a smooth \(N\)-dimensional domain \(\Omega\) with the condition that \(u\) tends to infinity when \(x\) tends to \(\partial\Omega\). We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal direction to 9!i. Motivated by the blow-up argument in our proof, we also give a symmetry result for some related problems in the half space.


35J60 Nonlinear elliptic equations
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[1] Porretta, Ve ron Symmetry properties of solutions of semilinear elliptic equa - tions in the plane Manuscripta, Math 115 pp 239– (2004)
[2] Diaz, Explosive solutions of quasilinear elliptic equations : existence and uniqueness Nonlinear, Anal 20 pp 97– (1993)
[3] Lions, Quelques remarques sur les problemes elliptiques quasilineaires du second ordre Analyse, Math 45 pp 234– (1985)
[4] Bandle, Large solutions of semilinear elliptic equations : existence uniqueness and asymptotic behaviour, Anal Math 58 pp 9– (1992) · Zbl 0802.35038
[5] Lasry, Nonlinear elliptic equations with singular boundary condi - tions and stochastic control with state constraints The model problem, Math Ann pp 283– (1989)
[6] Osserman, On the inequality u f u ) Pacific, Math 7 pp 1641– (1957)
[7] Marcus, Ve ron Existence and uniqueness results for large solutions of general nonlinear elliptic equations Evolution, Equ 3 pp 637– (2004)
[8] Bandle, Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Diff Eq 1 pp 133– (1996) · Zbl 0840.35034
[9] Marcus, Ve ron Uniqueness and asymptotic behaviour of solutions with boundary blow - up for a class of nonlinear elliptic equations, Inst 14 pp 237– (1997)
[10] Gilbarg, Partial Differential Equations of Second Order nd ed Springer Verlag Berlin New - York Ve ron On solutions of u u ) Pure, Appl Math 10 pp 378– (1983) · Zbl 0562.35001
[11] Bandle, Asymptotic behaviour of solutions and their derivatives for semilinear elliptic problems with blowup on the boundary Poincare Non Line aire, Inst Anal 12 pp 155– (1995)
[12] Lazer, Asymptotic behaviour of solutions of boundary blow up problems, Int Eq 7 pp 1001– (1994) · Zbl 0811.35010
[13] Diaz, Local estimates : uniqueness of solutions to some nonlinear elliptic equations Real, Rev Acad Cienc Exact Natur 88 pp 171– (1994)
[14] Ghergu, Explosive solutions of elliptic equations with absorption and non - linear gradient term Indian, Proc Acad Sci Math Sci 112 pp 441– (2002) · Zbl 1032.35070
[15] Porretta, Local estimates and large solutions for some elliptic equations with ab - sorption in, Diff Equ 9 pp 329– (2004) · Zbl 1150.35401
[16] Giarrusso, Asymptotic behaviour of large solutions of an elliptic quasilinear equation in a borderline case Paris, Acad Sci Ser Math pp 331– (2000) · Zbl 0966.35041
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