Blow‐up analysis, existence and qualitative properties of solutions for the two‐dimensional Emden–Fowler equation with singular potential

Motivated by the study of a two‐dimensional point vortex model, we analyse the following Emden–Fowler type problem with singular potential:$$ \left \{ \eqalign {&-\Delta u= \lambda {V(x){\rm{e}}^{u} \over {\int_{\Omega} V(x) {\rm{e}}^{u} {\rm d} x}} \quad {\rm in}\, \, \Omega \cr & u=0 \quad \quad \quad \quad \quad \quad\quad\quad\quad{\rm on}\, \, \Omega }\right.$$where V ( x ) = K ( x )/| x | 2α with α∈(0, 1), 0< a ⩽ K ( x )⩽ b < + ∞, ∀ x ∈Ω and ∥∇ K ∥ ∞ ⩽ C . We first extend various results, already known in case α⩽0, to cover the case α∈(0, 1). In particular, we study the concentration‐compactness problem and the mass quantization properties, obtaining some existence results. Then, by a special choice of K , we include the effect of the angular momentum in the system and obtain the existence of axially symmetric one peak non‐radial blow‐up solutions. Copyright © 2007 John Wiley & Sons, Ltd.