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

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