Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces
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| Format: | Preprint |
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2025
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| author | Gloss, Elisandra Mercuri, Carlo Perera, Kanishka Ribeiro, Bruno |
| author_facet | Gloss, Elisandra Mercuri, Carlo Perera, Kanishka Ribeiro, Bruno |
| contents | We prove existence and multiplicity results for the fractional Schroedinger--Poisson--Slater equation $(-Δ)^s u + (I_α* u^2)u = f(|x|,u)$ in $\mathbb{R}^N$, where $0<s<1$ and $α\in (1,N)$. 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 $\{λ_k\}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces Gloss, Elisandra Mercuri, Carlo Perera, Kanishka Ribeiro, Bruno Analysis of PDEs Functional Analysis 35R11, 35J60, 35A15, 35B33, 35J20 We prove existence and multiplicity results for the fractional Schroedinger--Poisson--Slater equation $(-Δ)^s u + (I_α* u^2)u = f(|x|,u)$ in $\mathbb{R}^N$, where $0<s<1$ and $α\in (1,N)$. 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 $\{λ_k\}$ 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. |
| title | Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces |
| topic | Analysis of PDEs Functional Analysis 35R11, 35J60, 35A15, 35B33, 35J20 |
| url | https://arxiv.org/abs/2511.04829 |