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.
We study the multiplicity of positive solutions of the critical elliptic equation: ∆ S 3 U = −(U 5 + λU ) on Ω that vanish on the boundary of Ω, where Ω is a region of S 3 which is invariant by the natural T 2 -action. H. Brezis and L. A. Peletier in [6] consider the case in which Ω is invariant by the SO(3)-action, namely, when Ω is a spherical cap. We show that the number of solutions increases as λ → −∞, giving an answer of a particular case of an open problem proposed by H. Brezis and L. A. Peletier in [6]. arXiv:1609.08011v2 [math.CA]
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