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

Abstract

We consider the following anisotropic Emden–Fowler equation

]where $\Omega \subset {\mathbb{R}}^2$ 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 $a\equiv \text{constant}$. We determine the localization and asymptotic behavior of the boundary bubbles, and construct some boundary bubbling solutions. In particular, we prove that if $\bar{x}\in \partial \Omega$ is a strict local minimum point of $a$, there exists a family of solutions such that ${\varepsilon }^{2}a(x)e^{u}dx$ tends to $8\pi a(\bar{x}){\delta }_{\bar{x}}$ in ${\mathcal{D}}^{\prime }({\mathbb{R}}^2)$ as $\varepsilon \to 0$. This result will enable us to get a new family of solutions for the isotropic problem $\mathrm{\Delta }u+{\varepsilon }^2{e}^u=0$ in rotational torus of dimension $N⩾3$.