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arXiv · 2511.04829

Fractional Schr\"odinger-Poisson-Slater equations in Coulomb-Sobolev spaces

Elisandra Gloss·Carlo Mercuri·Kanishka Perera·Bruno Ribeiro
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Abstract

We prove existence and multiplicity results for the fractional Schroedinger--Poisson--Slater equation $(-\Delta)^s u + (I_\alpha * u^2)u = f(|x|,u)$ in $\mathbb{R}^N$, where $0<s<1$ and $\alpha \in (1,N)$. We seek solutions in a fractional Coulomb-Sobolev space and employ new tools in critical point theory that link the behavior of $f$ at zero and at infinity to the scaling properties of the left-hand side. For several regimes of $f$, we establish compactness for an associated action functional and obtain multiple solutions as critical points, with the number governed by the interaction of $f$ with a sequence of eigenvalues $\{\lambda_k\}$ defined via the $\mathbb{Z}_2$ cohomological index of Fadell and Rabinowitz (rather than the classical Krasnosel'skii genus). In this fractional setting we also prove new regularity results and necessary conditions for the existence of solutions.

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Elisandra Gloss, Carlo Mercuri, Kanishka Perera, Bruno Ribeiro. 2025-11-06. Fractional Schr\"odinger-Poisson-Slater equations in Coulomb-Sobolev spaces. https://arxiv.org/abs/2511.04829

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