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

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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 -Δu+h(u)+ ∇ uq=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.