**Some Background:** A typical problem in mathematical physics is the existence of positive radially symmetric solutions to a nonlinear Schrodinger type equation over $\mathbb{R}^{2+1}$. Such a problem typically reduces to the following nonlinear boundary value problem (or "$n$-vortex equation" under the $n$-vortex ansatz $f(x)=f(r)e^{in\theta+i\lambda t}$),


\begin{align}
&f\_{rr}(r)+\dfrac{1}{r}f\_r(r)-\left(\dfrac{n^2}{r^2}+\lambda\right) f(r)+g(f^2(r))f(r)=0, \quad r\in(0,\infty),\label{vortexEq1}\\
&f(0)=0=f(\infty),
\end{align}
where $n\in\mathbb{Z}$, $\lambda>0$, and $g:\mathbb{R}\rightarrow\mathbb{R}$ is, say, a nice continuous function (e.g., $g(s)=s^2$ or $g(s)=s^2/(1+s^2)$). 


Via a variational approach, solutions to the above problem may be found as critical points of a corresponding functional, say, $I$, over an appropriate Sobolev space, say, $H$. However, a good understanding of the space considered is necessary (my current difficulty). 


**Question:** Let $H$ be the closure of $\mathcal{C}\_0^{\infty}(0,\infty)$ with inner product 
\begin{align}
(f,h)=\int\_0^{\infty}\left\{rf\_r(r)h\_r(r)+\left(\dfrac{n^2}{r^2}+\lambda\right) rf(r)h(r)\right\}dr,
\end{align}
and norm $||f||\_H^2=(f,f)$. Is $H$ compactly contained in $L^p((0,\infty),rdr)$ for all $p\geq 2$? (With particular interest in the case $p=2$.)


**Further background and results:** It is clear that $H$ is continuously embedded in $L^2((0,\infty),rdr)$ and $W^{1,2}((0,\infty),rdr)$, i.e., 
\begin{align}
||f||^2\_{L^2((0,\infty),rdr)}=\int\_0^{\infty}rf^2(r)dr\leq \int\_0^{\infty}\left\{f\_r^2(r)+f^2(r)\right\}rdr\leq\max\{1,1/\lambda\}||f||\_H^2.
\end{align}
We can also show that $f(0)=0$ and $f(\infty)=0$ for every $f\in H$.


In a particular problem, contained in the paper <https://doi.org/10.1515/jaa-2016-0010> by Carlo Greco published in the Journal of Applied Analysis, the author states that $H$ in compactly contained in $L^p((0,\infty),rdr)$ for all $p> 2$. Hence, my particular interest into the case $p=2$.



Is this space compactly contained in $L^p((0,\infty),rdr)$ for all $p\geq 2$?
The embedding is not compact this can be observed by considering the sequence 
$f\_k (r) = f (r/k)/k^2$ for some given function $f \in C^\infty\_c (0, +\infty) \setminus \{0\}$. One has for every $k \ge 1$, by a change of variable
$$
 \Vert f\_k \Vert^2 
= \frac{1}{k^2} \int\_0^\infty \vert f' (r)\vert^2 r + \frac{n \vert f (r)\vert^2}{r} \,\mathrm{d}r
+ \lambda \int\_0^\infty \vert f (r)\vert^2 r \,\mathrm{d}r
$$
Since the sequence converges to $0$ almost everywhere, its limit in $L^2$ can only be $0$ but 
$$
 \int\_0^\infty \vert f\_k (r) - 0 \vert^2 r \,\mathrm{d}r
= \int\_0^\infty \vert f (r) \vert^2 r \,\mathrm{d}r > 0,
$$
so that it cannot converge strongly to $0$.


