Open Access
Translator Disclaimer
1996 The scalar curvature equation on $\mathbb R^n$ and $S^n$
Gabriele Bianchi
Adv. Differential Equations 1(5): 857-880 (1996).


We study the existence of positive solutions for the equation $\Delta u+ K(x) u^{\frac{n+2}{n-2}}=0$ in $\mathbb{R}^n$ ($n \geq 3$) which decay to $0$ at infinity like $|x|^{2-n}$; $K(x)$ is a function which is bounded from above and below by positive constants, and no symmetry assumption on $K$ is made. We find conditions that guarantee existence for a large class of $K$'s. As a consequence one can explicitly show that the set of coefficients for which a solution exists is dense, in $C^1$ norm, in the space of positive bounded $C^1$ functions. These conditions also allow us to display a radial $K$ such that the previous problem has nonradial solutions but no radial solution. Some new results for the corresponding problem on $\Sn$ are also proved.


Download Citation

Gabriele Bianchi. "The scalar curvature equation on $\mathbb R^n$ and $S^n$." Adv. Differential Equations 1 (5) 857 - 880, 1996.


Published: 1996
First available in Project Euclid: 25 April 2013

zbMATH: 0865.35044
MathSciNet: MR1392008

Primary: 35J60
Secondary: 53C21 , 58G30

Rights: Copyright © 1996 Khayyam Publishing, Inc.


Vol.1 • No. 5 • 1996
Back to Top