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

TL;DR

This paper investigates the boundary behavior of the gradient of large solutions to certain nonlinear elliptic equations, revealing that blow-up phenomena primarily occur in the normal direction to the boundary.

Contribution

It provides precise asymptotic expressions for the gradient blow-up of solutions to nonlinear elliptic equations with boundary blow-up conditions.

Findings

Gradient blow-up occurs mainly in the normal direction.

Explicit asymptotic formulas for the gradient near the boundary.

A symmetry result for related problems in the half space.

Abstract

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 $-\Delta u+h(u)+\abs {\nabla u}^q=f$ 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 $\partial\Omega$. 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.