Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities

Abstract

We study a class of nonlinear nonlocal elliptic equations in ℝ^N involving superpositions of Hartree-type nonlinearities. Motivated by the Schrödinger-Poisson-Slater system, these equations arise as natural generalizations of problems with a single nonlocal interaction term. More precisely, we consider equations driven by a family of Riesz potentials weighted by a positive Borel measure, which gives rise to a superposed nonlocal operator. To treat this problem variationally, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb-Sobolev space, and study their main properties. Combining variational methods with a recently developed scaling-based critical point theory, we prove existence and multiplicity results for radial solutions. We also investigate a Brezis-Nirenberg-type problem and obtain multiplicity results near eigenvalues of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework for treating superpositions of nonlocal interactions of Hartree type.

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BibTeX RIS

BibTeX

@misc{marinho2026solutions,
  title = {{Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities}},
  author = {Artur Jorge Marinho and Kanishka Perera},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.05592},
  eprint = {2606.05592},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.05592},
}

RIS

TY  - UNPB
TI  - Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities
AU  - Artur Jorge Marinho
AU  - Kanishka Perera
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2606.05592
UR  - https://arxiv.org/abs/2606.05592
AB  - We study a class of nonlinear nonlocal elliptic equations in ℝ^N involving superpositions of Hartree-type nonlinearities. Motivated by the Schrödinger-Poisson-Slater system, these equations arise as natural generalizations of problems with a single nonlocal interaction term. More precisely, we consider equations driven by a family of Riesz potentials weighted by a positive Borel measure, which gives rise to a superposed nonlocal operator. To treat this problem variationally, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb-Sobolev space, and study their main properties. Combining variational methods with a recently developed scaling-based critical point theory, we prove existence and multiplicity results for radial solutions. We also investigate a Brezis-Nirenberg-type problem and obtain multiplicity results near eigenvalues of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework for treating superpositions of nonlocal interactions of Hartree type.
ER  -

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