Ground state solutions for a non-local type problem in fractional Orlicz Sobolev spaces

In this paper, we study the following nonlocal problem in fractional Orlicz Sobolev spaces \begin{eqnarray*} (-Δ_Φ)^{s}u+V(x)a(|u|)u=f(x,u),\quad x\in$\mathbb{R}$^N, \end{eqnarray*} where $(-{\mathrm{\Delta }}_{\mathrm{\Phi }}{)}^s(s\in (0,1))$ denotes the non-local and maybe non-homogeneous operator, the so-called fractional $\mathrm{\Phi }$-Laplacian. Without assuming the Ambrosetti-Rabinowitz type and the Nehari type conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case. The proof is based on a variant version of the mountain pass theorem and a Lions’ type result for fractional Orlicz Sobolev spaces.