Growth Estimates on Positive Solutions of the Equation

Leung, Man Chun

Canadian mathematical bulletin, Tome 44 (2001) no. 2, pp. 210-222

Résumé

We construct unbounded positive ${{C}^{2}}$ -solutions of the equation $\Delta u\,+\,K{{u}^{\left( n+2 \right)/\left( n-2 \right)}}\,=\,0$ in ${{\mathbb{R}}^{n}}$ (equipped with Euclidean metric ${{g}_{0}}$ ) such that $K$ is bounded between two positive numbers in ${{\mathbb{R}}^{n}}$ , the conformal metric $g\,=\,{{u}^{4/\left( n-2 \right)}}{{g}_{0}}$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$ , we obtain growth estimate on the ${{L}^{2n/\left( n-2 \right)}}$ -norm of the solution and show that it has slow decay.

Détail

DOI : 10.4153/CMB-2001-021-5

Mots-clés : 35J60, 58G03, positive solution, conformal scalar curvature equation, growth estimate