In this paper, we study the existence and concentration of positive solutions for the following fractional Schr\”odinger logarithmic equation: \begin{equation*} \left\{ \begin{aligned} & \varepsilon^{2s} (-\Delta)^{s} u+V(x)u =u\log u^2,\ x\in \mathbb{R}^N,\\ &u\in H^s(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $N>2s,$ $s \in ( 0 ,1), (-\Delta)^{s}$ is the fractional Laplacian, the potential $V$ is a continuous function having a global minimum. Using variational method to modify the nonlinearity with the sum of a $C^1 $ functional and a convex lower semicontinuous functional, we prove the existence of positive solutions and concentration around of a minimum point of $V$ when $\varepsilon$ tends to zero.

In this paper we consider the following fractional $(p,q)$-Laplacian equation $$ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right)=\lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $$ where $s \in(0,1), \lambda>0, 2