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

TL;DR

This paper establishes existence and multiplicity of solutions for a fractional Schrödinger-Poisson-Slater equation in Coulomb-Sobolev spaces, using novel critical point theory tools and eigenvalue analysis.

Contribution

It introduces new methods linking the behavior of the nonlinearity to the scaling properties of the fractional operator, and uses cohomological index to find multiple solutions.

Findings

Proved existence of solutions in fractional Coulomb-Sobolev spaces.

Established multiplicity results based on eigenvalues and cohomological index.

Derived new regularity conditions and necessary criteria for solutions.

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.