# Analysis of boundary bubbling solutions for an anisotropic Emden–Fowler equation

### Juncheng Wei

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong### Dong Ye

Département de Mathématiques, UMR 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France### Feng Zhou

Department of Mathematics, East China Normal University, Shanghai 200062, China

## Abstract

We consider the following anisotropic Emden–Fowler equation

]where $Ω⊂R_{2}$ is a smooth bounded domain and $a$ is a positive smooth function. We study here the phenomenon of boundary bubbling solutions which *do not exist* for the isotropic case $a≡constant$. We determine the localization and asymptotic behavior of the boundary bubbles, and construct some boundary bubbling solutions. In particular, we prove that if $xˉ∈∂Ω$ is a strict local minimum point of $a$, there exists a family of solutions such that $ɛ_{2}a(x)e_{u}dx$ tends to $8πa(xˉ)δ_{xˉ}$ in $D_{′}(R_{2})$ as $ɛ→0$. This result will enable us to get a new family of solutions for the isotropic problem $Δu+ɛ_{2}e_{u}=0$ in rotational torus of dimension $N⩾3$.

## Cite this article

Juncheng Wei, Dong Ye, Feng Zhou, Analysis of boundary bubbling solutions for an anisotropic Emden–Fowler equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 3, pp. 425–447

DOI 10.1016/J.ANIHPC.2007.02.001