Abstract:

For the following constrained minimization problem $ d_a(q)=\inf\limits_{\{u\in H^{\frac{1}{2}}(\mathbb{R}^3), \int_{\mathbb{R}^3}|u|^2{\rm d}x=1\}}{E_q(u)}, $ where $E_q(u)$ is the energy functional of the pseudo-relativistic Schrödinger equation $ E_q(u)=\frac{1}{2}\int_{\mathbb{R}^3}\bar{u}(\sqrt{-\Delta+m^2 }-m) u{\rm d}x-\frac{a}{q+2} \int_{\mathbb{R}^3}|u|^{q+2}{\rm d}x. $ For any $q\in(0, \frac{2}{3})$, the article proved that the problem (0.1) has at least one radially symmetric non-negative minimizer; and analyzed the blow-up behavior of the minimizer as $q\nearrow \frac{2}{3}$.

Key words: Ground states, Constrained variational problem, Existence, Blow-up