Finite Energy Solutions to the Yamabe Equation

We prove the existence of positive, finite energy solutions to the Yamabe equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta u - \frac{{n - 2}}{{4(n - 1)}}Su + u^{(n + 2)/(n - 2)} = 0$$ \end{document}on some noncompact manifolds with positive scalar curvature. We also clarify a published result on the existence of 'complete solutions' on those manifolds.