Analysis of boundary bubbling solutions for an anisotropic Emden-Fowler equation (Q930013)

Let us consider the anisotropic Emden-Fowler equation \[ \nabla (a(x)\nabla u )+ \varepsilon^2 a(x) e^u=0 \quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial \Omega, \] with \(\Omega \subset\mathbb R^n \) being a smooth bounded domain and \(a\) being a positive function. The authors study the phenomenon of boundary bubbling solutions which do not exist for the isotropic case \(a\) = constant. The localization and asymptotic behaviour of the boundary bubbles are determined. Further, the authors construct some boundary bubbling solutions. The basic idea is to consider axially symmetric solutions of the classical Emden-Fowler equation (or Gelfand equation) in a torus and try to give some precise descriptions of them.