Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations

If h is a nondecreasing real valued function and 0 ≤ q ≤ 2, we analyse the boundary behaviour of the gradient of any solution u of −∆u+ h(u) + |∇u|q = f in a smooth N-dimensional domain Ω with the condition that u tends to infinity when x tends to ∂Ω. 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 ∂Ω. Motivated by the blow–up argument in our proof, we also give in Appendix a symmetry result for some related problems in the half space. 1991 Mathematics Subject Classification. 35J60.