Multiple solutions for a class of fractional logarithmic Schrödinger equations

In this paper, we study the following fractional logarithmic Schrodinger equation: $$\begin{aligned} (-\Delta )^s u+V(x)u=u \log u^2,\quad x \in {\mathbb {R}}^{N}, \end{aligned}$$ where $$s\in (0,1)$$ , $$N>2s$$ , and the external potential V(x) belongs to $$C({\mathbb {R}}^N)$$ and is bounded from below. By the direction derivative and constrained minimization method, we obtain the existence of nonnegative and sign-changing weak solutions in $$H^s(R^N)$$ with various potentials. Moreover, we also construct a radial nodal solution, which changes sign exactly k-time (for any $$k \in {\mathbb {N}}$$ ) when the potential is radial symmetric.