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

We consider the following anisotropic Emden–Fowler equation∇(a(x)∇u)+ε2a(x)eu=0in Ω,u=0on ∂Ω, where Ω⊂R2Ω⊂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 x¯∈∂Ω is a strict local minimum point of a  , there exists a family of solutions such that ε2a(x)eudx tends to 8πa(x¯)δx¯ in D′(R2)D′(R2) as ε→0ε→0. This result will enable us to get a new family of solutions for the isotropic problem Δu+ε2eu=0Δu+ε2eu=0 in rotational torus of dimension N⩾3N⩾3.