Multipeak Solutions for the Yamabe Equation

Abstract

Let (M, g) be a closed Riemannian manifold of dimension $$n\ge 3$$ and $$x_0 \in M$$ be an isolated local minimum of the scalar curvature $$s_\mathrm{{g}}$$ of g. For any positive integer k we prove that for $$\epsilon >0$$ small enough the subcritical Yamabe equation $$-\epsilon ^2 \Delta u +(1+ c_{N} \epsilon ^2 s_\mathrm{{g}}) u = u^\mathrm{{q}}$$ has a positive k-peaks solution which concentrate around $$x_0$$ , assuming that a constant $$\beta $$ is non-zero. In the equation $$c_N = \frac{N-2}{4(N-1)}$$ for an integer $$N>n$$ and $$q= \frac{N+2}{N-2}$$ . The constant $$\beta $$ depends on n and N, and can be easily computed numerically, being negative in all cases considered. This provides solutions to the Yamabe equation on Riemannian products $$(M\times X , g+ \epsilon ^2 h )$$ , where (X, h) is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.