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 analyze the following Emden-Fowler type problem with singular potential \bgin{equation}\graf{ -\lapl u=\lm \dfrac{\e{u}}{\ino\e{u} \dx} & \mbox{in}\hspace{.2cm} \om, \nonumber\\\\ \hspace{.55cm}u=0 & \hspace{-.05cm} \mbox{on}\hspace{.2cm} \om, }\end{equation} where $\displaystyle V(x)=\frac{ K(x)}{|x|^{2\al}}$ with $\alpha\in(0,1)$, $0< a\leq K(x)\leq b<+\infty$, $\fal{x}{\om}$ and $\|\nabla K\|_\i\leq C$. We first extend various results, already known in case $\alpha\leq 0$, to cover the case $\alpha\in(0,1)$. 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.