## 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.

### MSC:

 35J60 Nonlinear elliptic equations
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### References:

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