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Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds

Bartosz Bieganowski, Pietro d'Avenia, Jacopo Schino

Abstract

We consider the singular elliptic problem of the form \[ -Δu + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-Δ+V$, geometric assumptions on the manifold $M$ ensuring the Sobolev embedding $H^1(M)\hookrightarrow L^{2^*}(M)$, and a suitable global integrability/smallness condition involving $\mathcal{A}$, $\mathcal{B}$, and a function $ψ\in H^1(M)$, we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $\mathcal{B}\ge 0$, we obtain a positive finite-energy solution. The proof relies on a family of $\varepsilon$-regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on $\mathcal{A}$ is, in a precise sense, necessary for the existence of nonnegative supersolutions.

Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds

Abstract

We consider the singular elliptic problem of the form where the coefficients are allowed to have low regularity. Under natural spectral assumptions on , geometric assumptions on the manifold ensuring the Sobolev embedding , and a suitable global integrability/smallness condition involving , , and a function , we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and , we obtain a positive finite-energy solution. The proof relies on a family of -regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on is, in a precise sense, necessary for the existence of nonnegative supersolutions.
Paper Structure (7 sections, 9 theorems, 86 equations)

This paper contains 7 sections, 9 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev spaces on manifolds
  4. Liouville-type theorems
  5. epsilon-approximations
  6. An alternative to (\ref{['eq:ThetaK']})
  7. Proof of the main results

Key Result

Theorem 1.2

Assume that the Ricci curvature of $(M,g)$ is bounded from below and If (AssA), (AssB), and (AssV) hold and there exist $K>0$, $\Theta >0$, and $\psi \in H^1 (M)$ such that and then eq:main admits a nonnegative finite-energy supersolution $u \in H^1 (M)$. If, moreover, the Ricci curvature of $(M,g)$ is nonnegative and $\mathcal{B} \ge 0$, then $u$ is a positive solution to eq:main.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7
  • Proposition 2.1: Harnack's inequality
  • proof
  • Remark 2.2
  • ...and 15 more