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Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

Bruno Bianchini, Luciano Mari, Marco Rigoli

Abstract

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a given function $\widetilde s(x)$ as its scalar curvature. In particular, the work focuses on the case when $\widetilde s(x)$ changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be quasi-isometric to the original one. The topological-geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of $M$. Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the type $$ Δ_{p} u + a(x)u^{p-1} - b(x)u^σ= 0 $$ where $Δ_p$ is the $p$-Laplacian, $σ>p-1>0$, $a,b \in L^\infty_{\mathrm{loc}}(M)$ and $b$ changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrödinger type operator $$ Q_V' \ : \ \varphi \longmapsto -Δ_p \varphi - a(x)|\varphi|^{p-2}\varphi. $$ In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

Abstract

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold , namely the existence of a conformal deformation of the metric realizing a given function as its scalar curvature. In particular, the work focuses on the case when changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be quasi-isometric to the original one. The topological-geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of . Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the type where is the -Laplacian, , and changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrödinger type operator In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

Paper Structure

This paper contains 10 sections, 40 theorems, 350 equations, 2 figures.

Table of Contents

  1. Introduction, I: existence for the generalized Yamabe problem
  2. Introduction, II: our main results in their general setting
  3. Preliminaries
  4. Criticality theory for $Q_V$, capacity and Hardy weights
  5. Hardy weights and comparison geometry
  6. Hardy weights on manifolds with a pole
  7. Hardy weights on minimally immersed submanifolds
  8. Specializing our main theorems: an example
  9. Proofs of Theorems \ref{['teouno']} and \ref{['teouno_bis']}
  10. Proofs of our geometric corollaries, and concluding comments

Key Result

Theorem 1.1

Let $(M^m, \langle \, , \, \rangle)$ be a complete manifold with dimension $m \ge 3$ and scalar curvature $s(x) \ge 0$. Suppose that $\mathrm{vol}(B_r(x)) \le Cr^m$ for some uniform $C$ independent of $x$, and that $M$ has positive Yamabe invariant $Y(M)$: $c_m$ as in 02. Assume further that Then, there exists a solution $u \in L^{\frac{2m}{m-2}}(M)$ of 02 such that for some $C>0$. In particula

Figures (2)

  • Figure 1: Euclidean space, \ref{['crescescaleugen']} and Zhang's assumptions on $\widetilde{s}(x)$ in force.
  • Figure 2: Manifolds close to $\mathbb{H}^m_\kappa$, $-C_1 \le \widetilde{s}(x) \le -C_2 < 0$ for large $x$.

Theorems & Definitions (104)

  • Theorem 1.1: zhang, Thm. 1.1
  • Remark 1.1
  • Theorem 1.2: WMninaitokawano
  • Theorem 1.3: avilesmcowen, Thm 4
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: bmr3, Thm 2
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.5
  • ...and 94 more