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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 existence and multiplicity of positive and sign changing solutions to the following Yamabe type equation $$ div g(a\nabla u) + bu = c|u|^{2*-2} u \quad\text{on } M $$ 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 $$ -\frac{4(m-1)}{m-2}\Delta_g u+R_g u = \kappa u^{2*-2}\quad\text{on } M. $$ 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; Dynamical systems and ergodic theory

video/mp4

Author affiliation: Universidad Nacional Autonoma de Mexico

2017-03-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61033

Unreviewed

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