Boosted Ground States for a Pseudo-Relativistic Schrödinger Equation with a double power nonlinearity

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $\psi (t,x)=e^{i\lambda t}\phi (x-vt)$ to the nonlinear pseudo-relativistic Schrödinger equation $$i{\mathrm{\partial }}_t\psi =(\sqrt{-\mathrm{\Delta }+m^2})\psi -\mathrm{\mid }\psi {\mathrm{\mid }}^{\frac{2}{N}}\psi -\mu \mathrm{\mid }\psi {\mathrm{\mid }}^q\psi \text{ on }{\mathbb{R}}^N,$$ for $m\ge 0$ and $\mathrm{\mid }v\mathrm{\mid }<1$. 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.