The present paper investigates the fractional relativistic Schrödinger equation involving critical growth and competing potentials: \( {\left\{ \begin{array}{ll} (-\Delta + m^2)^s u+\mathcal {Z}(\varepsilon x)u+ \lambda u=\mathcal {Q}(\varepsilon x)f(u)+|u|^{2_\sharp ^*-2}u, & x\in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^{2}dx=d^{2}, \end{array}\right. } \) where \((-\Delta + m^2)^s\) denotes the fractional relativistic Schrödinger operator with \(s \in (0,1)\) and \(\varepsilon >0\) is a small parameter, \(m,d > 0\) , \(N > 2s\) , \(2_\sharp ^*=\frac{2N}{N - 2s}\) is the critical exponent in the sense of the Sobolev embedding theorem, \(\lambda \) is an unknown parameter that appears as a Lagrange multiplier, and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a mass subcritical growth. Under appropriate assumptions on \(\mathcal {Z}\) and \(\mathcal {Q}\) , together with the minimization techniques and Ljusternik-Schnirelmann category theory, we derive the concentration behavior of positive normalized solutions to this kind of problem when \(\varepsilon > 0\) small enough, and establish the connection between the number of solutions and the potential profiles \(\mathcal {Z}\) and \(\mathcal {Q}\) . To some extent, our main theorems complement and extend the results of Ambrosio [4, 5], Sun et al. [51] and Zhang et al. [56].
