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arXiv · 2606.05592

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

Artur Jorge Marinho·Kanishka Perera
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Abstract

We study a class of nonlinear nonlocal elliptic equations in $\mathbb{R}^N$ involving superpositions of Hartree-type nonlinearities. Motivated by the Schr\"odinger-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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Artur Jorge Marinho, Kanishka Perera. 2026-06-04. Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities. https://arxiv.org/abs/2606.05592

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