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Blow‐up analysis, existence and qualitative properties of solutions for the two‐dimensional Emden–Fowler equation with singular potential
Author(s) -
Bartolucci Daniele,
Montefusco Eugenio
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.887
Subject(s) - mathematics , vortex , compact space , mathematical analysis , axial symmetry , cover (algebra) , type (biology) , quantization (signal processing) , angular momentum , singular point of a curve , mathematical physics , pure mathematics , geometry , classical mechanics , physics , thermodynamics , mechanical engineering , ecology , algorithm , biology , engineering
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.