Advances in Differential Equations

Asymptotic non-degeneracy of the multiple blow-up solutions to the Gel'fand problem in two space dimensions

Massimo Grossi, Hiroshi Ohtsuka, and Takashi Suzuki

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Abstract

We consider a sequence of solutions $u_n$ of the problem $$-\Delta u=\lambda e^u \quad \text{in $\Omega$}, \qquad u=0 \quad \text{on $\partial\Omega$},$$ with $\lambda=\{\lambda_n\}_{n\in\mathbb N}$ and blowing up at $m$ points $\kappa_1,\dots,\kappa_m$ in $\Omega$. Under some non-degeneracy assumption on some suitable finite-dimensional function (related to $\kappa_1,\dots,\kappa_m$) we show that $u_n$ is non-degenerate for $n$ large enough.

Article information

Source
Adv. Differential Equations Volume 16, Number 1/2 (2011), 145-164.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854333

Mathematical Reviews number (MathSciNet)
MR2766897

Zentralblatt MATH identifier
1227.35091

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35J08: Green's functions 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35Q35: PDEs in connection with fluid mechanics 76B47: Vortex flows 76F99: None of the above, but in this section

Citation

Grossi, Massimo; Ohtsuka, Hiroshi; Suzuki, Takashi. Asymptotic non-degeneracy of the multiple blow-up solutions to the Gel'fand problem in two space dimensions. Adv. Differential Equations 16 (2011), no. 1/2, 145--164. https://projecteuclid.org/euclid.ade/1355854333.


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