**Some Background:** A typical problem in mathematical physics is the existence of positive radially symmetric solutions to a nonlinear Schrodinger type equation. Such a problem, considered here over the full plane, typically reduces to the following nonlinear boundary value problem (or nonlinear eigenvalue problem)  
\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\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_rh_r+\left(\dfrac{n^2}{r^2}+\lambda\right) rfh\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 $p=2$ and why not for every $p\geq 1$.)

**Further background and results:** If is clear that $H$ is continuously embedded in $L^p((0,\infty),rdr)$ for all $p\geq 2$, i.e.,
\begin{align}
||f||^2_{L^p((0,\infty),rdr)}=\int_0^{\infty}rf^2dr\leq \int_0^{\infty}\left\{f_r^2+f^2\right\}rdr\leq\max\{1,1/\lambda\}||f||_H^2.
\end{align}
Further, it can be shown 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][1] 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 $p=2$. Also, why not for all $p\geq 1$.


  [1]: https://doi.org/10.1515/jaa-2016-0010