Ground states for fractional nonlocal equations with logarithmic nonlinearity

In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by $$\begin{cases}\mathcal{L}_{K} u(x)+u \log |u|+|u|^{q-2} u=0, & x \in \Omega \\ u=0, & x \in \mathbb{R}^{n} \backslash \Omega\end{cases}$$ where 2 < q < 2∗s, LK is a non-local operator, Ω is an open bounded set of Rn with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.