Existence and Limiting Profiles of Boosted Ground States for the Pseudo-Relativistic Schrödinger Equation with Focusing Power Type Nonlinearity

We study the following constrained minimization problem 0.1 \(\begin{aligned} d_{a,q}(1):=\inf _{\varphi \in \mathcal {S}_1}{E_{a,q}(\varphi )}, \end{aligned}\) $$\begin{array}{c}\hfill d_{a,q}(1):=\underset{\phi \in {\mathcal{S}}_1}{inf}{E}_{a,q}(\phi ),\end{array}$$ where the boosted energy functional \(E_{a,q}(\varphi )\) $E_{a,q}(\phi )$ is given by \(\begin{aligned} E_{a,q}(\varphi ):=\frac{1}{2}\int _{\mathbb {R}^3}\bar{\varphi }(\sqrt{-\Delta +m^2 }-m ) \varphi dx+\frac{i}{2}\int _{\mathbb {R}^3}\bar{\varphi }(v\cdot \nabla ) \varphi dx-\frac{a}{q+2} \int _{\mathbb {R}^3}|\varphi |^{q+2}dx, \end{aligned}\) $$\begin{array}{c}\hfill E_{a,q}(\phi ):=\frac{1}{2}{\int }_{{\mathbb{R}}^3}\overline{\phi }(\sqrt{-\mathrm{\Delta }+m^2}-m)\phi dx+\frac{i}{2}{\int }_{{\mathbb{R}}^3}\overline{\phi }(v·\mathrm{\nabla })\phi dx-\frac{a}{q+2}{\int }_{{\mathbb{R}}^3}{|\phi |}^{q+2}dx,\end{array}$$ and \(\varphi \) $\phi$ is a complex function, the parameters \(m,a>0\) $m,a>0$ , \(v\in \mathbb {R}^3\) $v\in {\mathbb{R}}^3$ with \(|v|<1\) $|v|<1$ , and the constraint \(\mathcal {S}_1\) ${\mathcal{S}}_1$ is defined as \(\begin{aligned} \mathcal {S}_1:=\left\{ \varphi \in H^{\frac{1}{2}}(\mathbb {R}^3):\int _{\mathbb {R}^3}|\varphi |^2dx=1\right\} . \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{S}}_1:=\left\{\phi ,\in ,H^{\frac{1}{2}},({\mathbb{R}}^3),:,{\int }_{{\mathbb{R}}^3},{|\phi |}^2,d,x,=,1\right\}.\end{array}$$ We prove that the problem Eq. 0.1 has at least one minimizer if \((a,q)\in \mathcal {D}:=(0,+\infty )\times (0,\frac{2}{3}]\setminus [a^{*},+\infty )\times \frac{2}{3}\) $(a,q)\in \mathcal{D}:=(0,+\infty )\times (0,\frac{2}{3}]\backslash [a_{}^{\ast },+\infty )\times \frac{2}{3}$ for some \(a^*>0\) $a^{\ast }>0$ , and there is no minimizer if \((a,q)\in (0,+\infty )\times (0,+\infty )\setminus \mathcal {D}\) $(a,q)\in (0,+\infty )\times (0,+\infty )\backslash \mathcal{D}$ . Moreover, we analyse the asymptotic behavior of minimizers as \((a,q)\in \mathcal {D}\rightarrow (a_0,q_0)\in \mathcal {D}\) $(a,q)\in \mathcal{D}\to (a_0,q_0)\in \mathcal{D}$ , and find that the minimizer of \(d_{a,q}(1)\) $d_{a,q}(1)$ converges to some minimizer of \(d_{a_0,q_0}(1)\) $d_{a_0,q_0}(1)$ in \(H^{\frac{1}{2}}(\mathbb {R}^3)\) $H^{\frac{1}{2}}({\mathbb{R}}^3)$ . In addition, when \((a,q)\in \mathcal {D}\rightarrow (\check{a},\frac{2}{3})\) $(a,q)\in \mathcal{D}\to (\check{a},\frac{2}{3})$ with \( \check{a}\in [a^{*},+\infty )\) $\check{a}\in [a_{}^{\ast },+\infty )$ , we show that all minimizers must blow up and present the detailed asymptotic behavior of minimizers.