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

Abstract

Abstract Motivated by the study of a two‐dimensional point vortex model, we analyse the following Emden–Fowler type problem with singular potential: 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.