Positive Solutions of Yamabe Type Equations on Complete Manifolds and Applications

✍ Scribed by L Brandolini; M Rigoli; A.G Setti

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 514 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3313

✦ Synopsis

We study the semilinear equation 2u+a(x) u=b(x) u _ (_>1) on a complete Riemannian manifold. We determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions. A very general uniqueness result is also established. Our main results are valid without explicit curvature assumptions, and appear to be new even in R m .

1998 Academic Press deform g to a (possibly non-complete) metric g~with scalar curvature K(x)?

The geometric problem is equivalent to the existence of a positive everywhere defined solution of an equation of the form (1.1), the scalar curvature equation (see, e.g., [Kz]): Indeed, setting g~=u 4Â(m&2) g, u>0, article no.