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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 boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish the existence and multiplicity of positive and sign changing solutions to the following Yamabe type equation $$ -div_g(a \nabla u) + bu = c|u|^{2^{*}-2} u \ \mbox{ on} \ M $$ where $div_g$ denotes the divergence operator on $(M; g)$, $a; b $ and $c$ are smooth functions with a and $c$ positive, and $2^{*}= \frac{2m}{m-2}$ denotes the critical Sobolev exponent. In particular, if $R_g$ denotes the scalar curvature, we give some examples where the Yamabe equation $$ -\frac{4 (m-1)}{m-2} \Delta_g u+ R_g u= \kappa u^{2^{*}-2} \ \mbox{on} \ M$$ 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 Monica Clapp.

Mathematics; Partial differential equations; Differential geometry

video/mp4

Author affiliation: Universidad Nacional Autonoma de Mexico

2017-11-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63687

Unreviewed

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