Uniform bound and a non-existence result for Lichnerowicz equation in the whole \(n\)-space. (English) Zbl 1167.35320
Summary: We give a uniform bound and a non-existence result for positive solutions to the Lichnerowicz equation in \(\mathbb{R}^n\).
In particular, we show that positive smooth solutions to \(\Delta u+f(u)=0,\quad u>0,\quad \text{ in }\mathbb{R}^n\) where \(f(u)=u ^{- p - 1} - u^{p - 1},\) are uniformly bounded.
In particular, we show that positive smooth solutions to \(\Delta u+f(u)=0,\quad u>0,\quad \text{ in }\mathbb{R}^n\) where \(f(u)=u ^{- p - 1} - u^{p - 1},\) are uniformly bounded.
References:
| [1] | Aubin, Th., Some Nonlinear Problems in Riemannian Geometry (1998), Springer: Springer New York · Zbl 0896.53003 |
| [2] | Choquet-Bruhat, Y.; Isenberg, J.; Pollack, D., The Einstein-scalar field constraints on asymptotically Euclidean manifolds, Chinese Ann. Math. Ser. B, 27, 1, 31-52 (2006) · Zbl 1112.83008 |
| [3] | Choquet-Bruhat, Y.; Isenberg, J.; Pollack, D., The constraint equations for the Einstein-scalar field system on compact manifolds, Class. Quantum Grav., 24, 809-828 (2007) · Zbl 1111.83002 |
| [4] | O. Druet, E. Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on a compact Riemannian manifolds, 2008, preprint; O. Druet, E. Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on a compact Riemannian manifolds, 2008, preprint · Zbl 1182.83002 |
| [5] | Du, Y.; Ma, L., Logistic type equations on \(R^N\) by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64, 107-124 (2001), MR 2002d:35089 · Zbl 1018.35045 |
| [6] | Hebey, E.; Pacard, F.; Pollack, D., A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Comm. Math. Phys., 278, 1, 117-132 (2008) · Zbl 1218.53042 |
| [7] | Lee, J.; Parker, Th., The Yamabe problem, Bull. Am. Soc., 17, 1, 37-91 (1987) · Zbl 0633.53062 |
| [8] | R. Schoen, A report on some recent progress on nonlinear problems in differential geometry, Surveys in Differential Geometry, 1991, pp. 201-241; R. Schoen, A report on some recent progress on nonlinear problems in differential geometry, Surveys in Differential Geometry, 1991, pp. 201-241 · Zbl 0752.53025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
