Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds
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arXiv
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2026
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| author | Bieganowski, Bartosz d'Avenia, Pietro Schino, Jacopo |
| author_facet | Bieganowski, Bartosz d'Avenia, Pietro Schino, Jacopo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09889 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds Bieganowski, Bartosz d'Avenia, Pietro Schino, Jacopo Analysis of PDEs Mathematical Physics Differential Geometry 58J05, 58J90, 35J75, 35J61, 35Q75, 35Q76 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. |
| title | Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry 58J05, 58J90, 35J75, 35J61, 35Q75, 35Q76 |
| url | https://arxiv.org/abs/2603.09889 |