Bubbling solutions for an anisotropic Emden-Fowler equation

We consider the 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 (x) is a positive, smooth function. We investigate the effect of anisotropic coefficient a (x) on the existence of bubbling solutions. We show that at given strict local maximum points of a, there exist solutions with arbitrarily many bubbles. As a consequence, the quantityT_ε = ε² under(∫, Ω) a (x) e^u d x can approach to +∞ as ε → 0. These results show a striking difference with the isotropic case (a (x) ≡ constant). To cite this article: J. Wei et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).