Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3

Abstract

<p style='text-indent:20px;'>On a closed <inline-formula><tex-math id="M2">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-dimensional Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> we investigate the limit of the Einstein-Lichnerowicz equation</p><p style='text-indent:20px;'><disp-formula> <label>1</label> <tex-math id="E1"> \begin{document}$ \begin{equation} \triangle_g u + h u = f u^5 + \frac{\theta a}{u^7} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>as the momentum parameter <inline-formula><tex-math id="M4">\begin{document}$ \theta \to 0 $\end{document}</tex-math></inline-formula>. Under a positive mass assumption on <inline-formula><tex-math id="M5">\begin{document}$ \triangle_g +h $\end{document}</tex-math></inline-formula>, we prove that sequences of positive solutions to this equation converge in <inline-formula><tex-math id="M6">\begin{document}$ C^2(M) $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M7">\begin{document}$ \theta \to 0 $\end{document}</tex-math></inline-formula>, either to zero or to a positive solution of the limiting equation <inline-formula><tex-math id="M8">\begin{document}$ \triangle_g u + h u = f u^5 $\end{document}</tex-math></inline-formula>. We also prove that the minimizing solution of (1) constructed by the author in [<xref ref-type="bibr" rid="b15">15</xref>] converges uniformly to zero as <inline-formula><tex-math id="M9">\begin{document}$ \theta \to 0 $\end{document}</tex-math></inline-formula>.</p>