**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 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$ .