Multipeak Solutions for the Yamabe Equation

Abstract

Let (M, g) be a closed Riemannian manifold of dimension n ≥ 3 and x 0 ∈ M be an isolated local minimum of the scalar curvature s g of g. For any positive integer k we prove that for ǫ > 0 small enough the subcritical Yamabe equation −ǫ 2 ∆u + (1 + c N ǫ 2 s g )u = u q has a positive k-peaks solution which concentrate around x 0 , assuming that a constant β is non-zero. In the equation c N = N −2 4(N −1)for an integer N > n and q = N +2 N −2 . The constant β 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 × X, g + ǫ 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.

A generic multiplicity result for a Yamabe-type equation via Morse theory

“…This exact same setting was studied by J. Petean in [Pet19] and by C. Rey and J. Ruiz in [RR18]. In [Pet19], J. Petean proves that the Yamabe problem in our context admits at least Cat(M ) + 1 solutions for small enough > 0, where Cat(M ) is the Lusternik-Schnirelmann category of M .…”

“…For instance, multiplicity of solutions for the Yamabe problem in such setting was the object of study of R. Bettiol and P. Piccione in [BP13]. Closer to our context, the exact same setting which we consider was studied by J. Petean in [Pet19] and by C. Rey and J. Ruiz in [RR18].…”

“…In [Pet19], J. Petean explicits a connection between the number of solutions of eq. In [RR18], C. Rey and J. Ruiz build k-peak solutions for eq. (5.3) concentrating around x 0 ∈ M an isolated local minimum of scal(g) for small > 0 and any positive integer k.…”

“…More recently, J. N. A. Neto generalized this result to the context of Riemannian submersions with minimal fibers in [dAN20]. In [RR18], C. Rey and J. Ruiz build k-peak solutions for the Yamabe problem concentrating around minima of scal(g).…”