| Abstract: |
| We study a class of nonlinear nonlocal elliptic equations in $R^N$ involving a superposition of Hartree-type nonlinearities. These equations are motivated by the Schr\odinger--Poisson--Slater system and arise as natural generalizations of problems with a single nonlocal interaction term.
We consider an equation driven by a family of Riesz potentials weighted by a positive Borel measure, leading to a superposed nonlocal operator. To treat this problem, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb--Sobolev space, and investigate their properties.
Using variational methods combined with a recently developed scaling-based critical point theory, we establish existence and multiplicity results for radial solutions for a Brezis--Nirenberg type problem and show that multiplicity depends on the spectral properties of an associated nonlinear eigenvalue problem.
Our results extend previous works on single Hartree-type equations and provide a unified framework to handle superpositions of nonlocal interactions of this type. |
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