Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

TL;DR

This work extends the fractional Schrödinger–Poisson–Slater framework to the Coulomb–Sobolev setting, proving existence and multiplicity results for subcritical and critical nonlinearities in $\mathbb{R}^N$ with $0<s<1$ and $\alpha\in(1,N)$. It builds a robust variational structure on the fractional Coulomb–Sobolev space, leveraging a scaling analysis $u_t=t^{\theta}u(tx)$ to define a nonlinear eigenvalue sequence $\{\lambda_k\}$ via the $\mathbb{Z}_2$ cohomological index, rather than Krasnosel'skii genus. The paper establishes compactness and PS conditions in subcritical regimes, derives PS-conditions below the Sobolev-type threshold for critical problems, and uses critical-point theory with a pseudo-index to obtain multiple pairs of nontrivial solutions, including under near-eigenvalue resonances and in subscaled/anisotropic regimes. Regularity and Pohozaev-type identities are developed in the fractional nonlocal setting, and a density result for smooth functions in the fractional Coulomb–Sobolev space underpins the analysis. Overall, the results significantly extend existing $s=1$ Schrödinger–Poisson–Slater theory to the fractional case, providing a comprehensive multiplicity landscape for nonlocal Coulomb interactions.

Abstract

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.

Theorems & Definitions (57)

• Remark 1.1
• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9