## Differential and Integral Equations

### Asymptotics for some quasilinear elliptic equations

Emmanuel Hebey

#### Abstract

Let $B$ be the unit ball of $\Bbb R^N$, $N \ge 3$. We consider the problem $\Delta u = f(|x|)u^{p-\varepsilon}$ in $B$, $u > 0$ in $B$, $u = 0$ on $\partial B$, where $f\in C^\infty(\Bbb R, \Bbb R)$, $p = (N+2)/(N-2)$, $\varepsilon\ge 0$. First, we study the behavior of the minimizing radially symmetric solutions of the problem when $\varepsilon\to 0^+$. According to the (local or global) monotony of $f$, they converge to a solution of the critical problem or they blow up. The two cases are described. As a consequence, for large $N$ and all $\varepsilon > 0$, the critical problems with $f(r) = 1+\varepsilon r^k$ have minimizing radially symmetric solutions. They necessarily blow up when $\varepsilon\to 0^+$. Here again, we describe the blow up.

#### Article information

Source
Differential Integral Equations, Volume 9, Number 1 (1996), 71-88.

Dates
First available in Project Euclid: 7 May 2013

https://projecteuclid.org/euclid.die/1367969989

Mathematical Reviews number (MathSciNet)
MR1364035

Zentralblatt MATH identifier
0842.35034

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Hebey, Emmanuel. Asymptotics for some quasilinear elliptic equations. Differential Integral Equations 9 (1996), no. 1, 71--88. https://projecteuclid.org/euclid.die/1367969989