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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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