Fractional Schrödinger-Poisson-Slater Equations in Coulomb-Sobolev Spaces

We prove existence and multiplicity results for the fractional Schrödinger–Poisson–Slater equation (Formula presented.) with 0<1, α∈(1,N). We seek solutions in a fractional Coulomb–Sobolev space by employing new tools in critical point theory, which connect the behavior of the nonlinearity f at zero and at infinity with the scaling properties of the left hand side of the equation. In various regimes for the nonlinearity f, we establish compactness results for an associated action functional and find multiple solutions as critical points, whose number is sensitive to the interaction of f with a sequence of eigenvalues {λk} defined via the Z2–cohomological index of Fadell and Rabinowitz. The use of this index, instead of the classical Krasnoselskii genus, is essential for us to use new critical group estimates and scaling-based linking geometries. In this fractional setting, new regularity results and necessary conditions for solutions to exist are also proved.