In this paper, we consider the following fractional Schrödinger equation with critical and logarithmic nonlinearities: \(\begin{aligned} \left\{ \begin{array}{lll} \varepsilon ^{sp}(-\Delta )^{s}_{p}u+\mathcal V(x)|u|^{p-2}u=|u|^{p-2}u\log |u|^{p}+|u|^{p^{*}_{s}-2}u & {\text { in }} \mathbb {R}^{N},\\ u\in \mathcal W^{s,p}(\mathbb {R}^N),\ u > 0 & {\text { in }} \mathbb {R}^{N}, \end{array} \right. \end{aligned}\) where \(\varepsilon> 0, s\in (0,1), p\in (1,\infty ), N > sp, (-\Delta )_{p}^{s}\) is the fractional p-Laplacian, \(\mathcal {V}:\mathbb {R}^{N}\rightarrow \mathbb {R}\) is a continuous potential. By variational arguments and concentration compactness principle, we obtain the existence and multiplicity of positive solutions. Moreover, we also obtain the concentration and decay of solutions when the parameter \(\varepsilon \) is small enough. The feature and novelty of this paper lies in the fact that the above equation simultaneously contains both critical and logarithmic nonlinearities. To some extent, we extended the results of Alves and Ambrosio (Anal Appl 22:311–349, 2024), Alves and Ji (Calc Var Partial Differ Equ 59:21, 2024) and Ambrosio and Isernia (Discrete Contin Dyn Syst 38:5835–5881, 2018).
