## Differential and Integral Equations

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

### Asymptotics for some quasilinear elliptic equations

#### 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

**Permanent link to this document**

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