Boosted Ground States for a Pseudo-Relativistic Schr\"odinger Equation with a double power nonlinearity

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $\psi(t,x)=e^{i\lambda t}\varphi\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schr\"odinger equation $$i\partial_t \psi=(\sqrt{-\Delta+m^2})\psi - |\psi|^{\frac{2}{N}}\psi-\mu|\psi|^{q}\psi~~\text{ on }\mathbb{R}^N,$$ for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $\lambda$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.