Perturbation limiting behaviors of normalized ground states to focusing mass-critical Hartree equations with Local repulsion

Abstract

In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential

$$\begin{aligned} i\partial _t\psi =-\Delta \psi +|x|^2\psi -(|x|^{-2}*|\psi |^2) \psi +\varepsilon |\psi |^{p-2}\psi ,\ \ \text {in}\ \mathbb {R}^+ \times \mathbb {R}^N, \end{aligned}$$

where \(N\ge 3\), \(2<p<2^*={2N}/({N-2})\) and \(\varepsilon >0\). We mainly focus on the normalized ground state solitary waves of the form \(\psi (t,x)=e^{i\mu t}u_{\varepsilon ,\rho }(x)\), where \(u_{\varepsilon ,\rho }(x)\) is radially symmetric-decreasing and \(\int _{\mathbb {R}^N}|u_{\varepsilon ,\rho }|^2\,dx=\rho \). Firstly, we prove the existence and nonexistence of normalized ground states under the \(L^2\)-subcritical, \(L^2\)-critical (\(p=4/N +2\)) and \(L^2\)-supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states \(u_{\varepsilon ,\rho }\) as \(\varepsilon \rightarrow 0^+\) and find that the \(\varepsilon \)-blow-up phenomenon happens for \(\rho \ge \rho _c=\Vert Q\Vert ^2_{L^2}\), where Q is a positive radially symmetric ground state of \(-\Delta u+u-(|x|^{-2}*|u|^2)u=0\) in \(\mathbb {R}^N\). We prove that \(\int _{\mathbb {R}^N}|\nabla u_{\varepsilon ,\rho }(x)|^2\,dx\sim \varepsilon ^{-\frac{4}{N(p-2)+4}}\) for \(\rho =\rho _c\) and \(2<p<2^*\), while \(\int _{\mathbb {R}^N}|\nabla u_{\varepsilon ,\rho }|^2\,dx\sim \varepsilon ^{-\frac{4}{N(p-2)-4}}\) for \(\rho >\rho _c\) and \(4/N+2<p<2^*\), and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as \(\varepsilon \rightarrow +\infty \), which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer \(u^{TF}=\left[ \mu ^{TF}-|x|^2 \right] ^{\frac{1}{p-2}}_{+}\), where \(\mu ^{TF}\) is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for \(u_{\varepsilon , \rho }\) that \(\Vert u_{\varepsilon , \rho }\Vert _{L^{\infty }}\sim \varepsilon ^{-\frac{N}{N(p-2)+4}}\) as \(\varepsilon \rightarrow +\infty \).