Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential

In this article we consider the following fractional semilinear elliptic equation \[(-Δ)^su+|x|^2u =ω u+|u|2σu in R^N,\] where \(s∈ (0,1)\), \(N> 2s\), \(σ∈ (0,2s/N-2s)\) and \(ω∈ (0, λ_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result.