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

We consider the following anisotropic Emden-Fowler equation∇ (a (x) ∇ u) + ε² a (x) e^u = 0 in Ω, u = 0 on ∂ Ω, where Ω ⊂ R² 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 over(x, ̄) ∈ ∂ Ω is a strict local minimum point of a, there exists a family of solutions such that ε² a (x) e^u d x tends to 8 π a (over(x, ̄)) δ_{over(x, ̄)} in D^′ (R²) as ε → 0. This result will enable us to get a new family of solutions for the isotropic problem Δ u + ε² e^u = 0 in rotational torus of dimension N ≥ 3.