Estimates for explosive solutions to \(p\)-Laplace equations. (English) Zbl 0916.35041

Amann, H. (ed.) et al., Progress in partial differential equations. Papers from the 3rd European conference on elliptic and parabolic problems, Pont-à-Mousson, France, June 1997. Vol. 1. Harlow: Longman. Pitman Res. Notes Math. Ser. 383, 117-127 (1998).
We deal with the following problem: \[ \text{div}(|\nabla u|^{p- 2}\nabla u)= f(u)\quad\text{in }D,\quad u\to\infty\quad\text{as }x\to\partial D, \] where \(p>1\), \(f(t): (t_0,\infty)\to \mathbb{R}^+\) is smooth, increasing and satisfies the condition \[ \psi(t):= \int^\infty_t {ds\over (qF(s))^{1/p}}<\infty,\quad F(s)= \int^s_{t_0} f(t)dt,\quad q= {p\over p-1}. \] We prove that if \(F(t)t^{-p}\) is increasing for large \(t\) then a solution \(u(x)\) to this problem satisfies \[ | u(x)- \phi(\delta(x))|\leq c\delta(x)\phi(\delta(x)), \] where \(\phi\) is the inverse of the function \(\psi\) and \(\delta\) is the distance to the boundary. If in addition, \(F(t)t^{-2p}\to \infty\) as \(t\to\infty\) then also \(\lim_{x\to\partial D}(u(x)- \phi(\delta(x)))= 0\) holds.
For the entire collection see [Zbl 0905.00059].


35J67 Boundary values of solutions to elliptic equations and elliptic systems
35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations