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Abstract In the supercritical range of the polytropic indices$$\gamma \in (1,\frac{4}{3})$$ we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case$$\gamma =1$$ . They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.
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This content will become publicly available on February 15, 2026
Prescribing $Q$-curvature on even-dimensional manifolds with conical singularities
On a2m-dimensional closed manifold, we investigate the existence of prescribedQ-curvature metrics with conical singularities. We present here a general existence and multiplicity result in the supercritical regime. To this end, we first carry out a blow-up analysis of a2mth-order PDE associated to the problem, and then apply a variational argument of min-max type. Form>1, this seems to be the first existence result for supercritical conic manifolds different from the sphere.
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- Award ID(s):
- 2154219
- PAR ID:
- 10610920
- Publisher / Repository:
- EMS
- Date Published:
- Journal Name:
- Revista Matemática Iberoamericana
- Volume:
- 41
- Issue:
- 1
- ISSN:
- 0213-2230
- Page Range / eLocation ID:
- 1 to 28
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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