## 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].

### MSC:

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

### Keywords:

$$p$$-Laplacian; blow-up at the boundary