Global bifurcation techniques for Yamabe type equations on Riemannian\n manifolds

2019/05/22 by Alejandro Betancourt de la Parra, de la Parra, Alejandro Betancourt, Jurgen Julio-Batalla +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1905.09305

openalex publication_date 2019/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a closed Riemannian manifold (Mⁿ ,g) of dimension n\≥ 3\nand study positive solutions of the equation -\Δ_g u + \λ u = \λν^q, with \λ >0, q>1. If M supports a proper isoparametric function\nwith focal varieties M₁, M₂ of dimension d₁ \≥ d₂ we show that for\nany q<\( n-d₂+2 )/(n - d₂ -2) the number of positive solutions of the\nequation -\Δ_g u + \λ u = \λ u^q tends to \∞ as \λ\n\→ +\∞. We apply this result to prove multiplicity results for\npositive solutions of critical and supercritical equations. In particular we\nprove multiplicity results for the Yamabe equation on Riemannian manifolds.\n

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