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MULTIPLICITY OF NODAL SOLUTIONS FOR YAMABE TYPE EQUATIONS

Banff International Research Station for Mathematical Innovation and Discovery

Given a compact Riemannian manifold (M, g) without bound- ary of dimension m ≥ 3 and under some symmetry assumptions, we estab- lish existence and multiplicity of positive and sign changing solutions to the following Yamabe type equation −divg(a∇u) + bu = c|u|2∗−2u on M where divg denotes the divergence operator on (M, g), a, b and c are smooth functions with a and c positive, and 2∗ = 2m denotes the critical Sobolev m−2 exponent. In particular, if Rg denotes the scalar curvature, we give some examples where the Yamabe equation −4(m − 1)∆gu + Rgu = κu2∗−2 on M. m−2 admits an infinite number of sign changing solutions. We also study the lack of compactness of these problems in a symmetric setting and how the symmetries restore it at some energy levels. This allows us to use a suitable variational principle to show the existence and multiplicity of such solutions. This is joint work with M ́onica Clapp.

Mathematics; Geometry; Partial differential equations; Differential geometry

video/mp4

Author affiliation: Universidad Nacional Autonoma de Mexico

2017-06-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61980

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/