TITLE

Bubbling solutions for an anisotropic Emden—Fowler equation

AUTHOR(S)
Juncheng Wei; Dong Ye; Feng Zhou
PUB. DATE
February 2007
SOURCE
Calculus of Variations & Partial Differential Equations;Feb2007, Vol. 28 Issue 2, p217
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We consider the following anisotropic Emden–Fowler equation $$\nabla (a(x) \nabla u)+ \epsilon^{2} a(x) e^{u} = 0 \quad in \quad \Omega, \quad u=0 \quad on \quad \partial \Omega,$$ where $$\Omega \subset \mathbb{R}^2$$ is a bounded smooth domain and a( x) is a positive smooth function. We investigate the effect of anisotropic coefficient a( x) on the existence of bubbling solutions. We show that at given local maximum points of a( x), there exists arbitrarily many bubbles. As a consequence, the quantity $$\mathcal{T}_\epsilon = \epsilon^{2} \int_{\Omega} a(x)e^{u} {\rm d}x $$ can approach to $$ + \infty$$ as $$\epsilon \to 0$$ . These results show a striking difference with the isotropic case [ $$a(x) \equiv $$ Constant].
ACCESSION #
23218187

 

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