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1996 Asymptotics for some quasilinear elliptic equations
Emmanuel Hebey
Differential Integral Equations 9(1): 71-88 (1996).

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.

Citation

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Emmanuel Hebey. "Asymptotics for some quasilinear elliptic equations." Differential Integral Equations 9 (1) 71 - 88, 1996.

Information

Published: 1996
First available in Project Euclid: 7 May 2013

zbMATH: 0842.35034
MathSciNet: MR1364035

Subjects:
Primary: 35J60
Secondary: 35B40

Rights: Copyright © 1996 Khayyam Publishing, Inc.

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Vol.9 • No. 1 • 1996
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