Blowing-up solutions to an anisotropic Emden–Fowler equation with a singular source

Abstract

We consider the following anisotropic Emden–Fowler equation with a singular source − div ( a ( x ) ∇ v ) = ε 2 a ( x ) e v − 4 π α a ( p ) δ p in Ω , v = 0 on ∂ Ω , where p ∈ Ω ⊂ R 2 , constant α ∈ ( 0 , ∞ ) ∖ N , a ( x ) is a positive smooth function and δ p denotes the Dirac measure with pole at point p. If p is a local maximum point of a ( x ) , we construct a family of solutions v ε with arbitrary m bubbles concentrating at p, and the quantity ε 2 ∫ Ω a ( x ) e v ε → 8 π ( m + 1 + α ) a ( p ) .