2$, $\sum t_n^{2/q}=1$ only if all but one of $t_n$, say for $n=n_0$, equals zero. We conclude that $w^{(n_0)}$ is the minimizer for (\ref{theproblem}). \end{proof} \begin{remark} \rm We note that from the proof of Theorem~\ref{main} follows that that for any minimizing sequence $u_k$ for (\ref{theproblem}) there is a sequence $\eta_k$, such that $g_{\eta_k}u_k$ converges to the minimizer in $H^1_\alpha(M)$. Indeed, with $\eta_k=(\eta^{n_0}_k)^{-1}$ as above we have a weak convergence and convergence of the norms, and thus the norm convergence. \end{remark} \begin{thebibliography}{00} \bibitem{AHS} Avron J., Herbst I., Simon B.; \emph{Schr\"odinger operators with magnetic fields. I. General Interactions}, Duke Math. J. {\bf 45}, 847--883 (1978). \bibitem{BiSchiTi} Biroli M., Schindler I., Tintarev K.; \emph{Semilinear equations on Hausdorff spaces with symmetries}, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) {\bf 27}, 175-189 (2003). \bibitem{BrezisLieb} Brezis H., Lieb E.; \emph{A relation between pointwise convergence of functions and convergence of functionals}, Proc. Amer. Math. Soc. {\bf 88}, 486-490 \bibitem{Danielli} Danielli D.; \emph{A compact embedding theorem for a class of degenerate Sobolev spaces}, Rend. Sem. Mat. Univ. Politec. Torino {\bf 49}, 399-420 (1991). \bibitem{EstebanLions} Esteban, M. J., Lions, P.-L.; \emph{Stationary solutions of nonlinear Schr\"odinger equations with an external magnetic field}, Partial differential equations and the calculus of variations, Vol. I, 401-449, Progr. Nonlinear Differential Equations Appl., 1, Birkh\"auser Boston, Boston, MA, 1989. \bibitem{FiesTin} Fieseler K.-H., Tintarev K.; \emph{Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds}, J. Geom. Anal, {\bf 13}, 67-75 (2003). \bibitem{GilbTrud} Gilbarg D.,Trudinger N.S.; \emph{Elliptic Partial Differential Equations of Second Order}, 2nd ed., Springer Verlag, 1983. \bibitem{FollandStein} Folland, G. B.; Stein, E. M.; \emph{Estimates for the $\bar \partial \sb{b}$ complex and analysis on the Heisenberg group}. Comm. Pure Appl. Math. {\bf 27}, 429-522 (1974). \bibitem{Folland} Folland, G. B.; \emph{Subelliptic estimates and function spaces on nilpotent Lie groups}, Ark. Math. {\bf 13}, 161-207 (1975). \bibitem{Gruber} Gruber, M. J., \emph{Bloch theory and quantization of magnetic systems}. J. Geom. Phys. {\bf 34}, 137-15(2000). \bibitem{Kob} Kobayashi, S.; \emph{Transformation groups in differential geometry}, Ergebnisse der Matematik und ihren Grenzgebiete {\bf 70} (1992) \bibitem{Kurata} Kurata K.; \emph{Existence and semiclassical limit of the least energy solution to a nonlinear Schr\"odinger equation with electromagnetic fields}, Nonlinear Anal. {\bf 41} Ser. A: Theory Methods, 763-778 (2000). \bibitem{LL} Lieb E. H., Loss M.; \emph{Analysis}. Second edition. Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. \bibitem{PLL1a} Lions P.-L.; \emph{The concentration-compactness principle in the calculus of variations. The locally compact case, part 1}. Ann. Inst. H. Poincare, Analyse non lin\'eaire {\bf 1}, 109-1453 (1984). \bibitem{PLL1b} Lions P.-L.; \emph{The concentration-compactness principle in the calculus of variations. The locally compact case, part 2}. Ann. Inst. H. Poincare, Analyse non lin\'eaire {\bf 1}, 223-283 (1984). \bibitem{Lions86} Lions P.-L.; \emph{Solutions of Hartree-Fock equations for Coulomb systems}, Comm. Math. Phys. {\bf 109}, 33-97 (1987). \bibitem{acc} Schindler I., Tintarev K.; \emph{An abstract version of the concentration compactness principle}, Revista Matematica Complutense {\bf 15}, 417-436 (2002) \bibitem{Struwe} Struwe M.; \emph{Variational Methods}, Springer-Verlag 1990. \bibitem{Varopoulos} Varopoulos, N. Th.; \emph{Analysis on Lie Groups}, J. Func. Anal {\bf 76}, 346-410 (1988). \end{thebibliography} \end{document}