Limit Behavior of Mass Critical Hartree Minimization Problems with Steep Potential Wells

Abstract

We consider minimizers of the following mass critical Hartree minimization problem: \[ e_λ(N):=\underset{\{u\in H^1(R^d),\,\|u\|^2_2=N\}}{\inf} E_λ(u),\,\ d\ge 3, \] where the Hartree energy functional $E_{\lambda }(u)$ is defined by \[ E_λ(u):=\int_{R ^d}|\nabla u(x)|^2dx+λ\int_{R ^d}g(x)u^2(x)dx-\frac{1}{2} \int_{R ^d}\int_{R ^d} \frac{u^2(x)u^2(y)}{|x-y|^2}dxdy,\,\ λ>0,\] and the steep potential $g(x)$ satisfies $0=g(0)={\inf }_{R^d}g(x)\le g(x)\le 1$ and $1-g(x)\in L^{\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^{\ast }>0$, independent of $\lambda g(x)$, such that if $N\ge N^{\ast }$, then $e_{\lambda }(N)$ does not admit minimizers for any $\lambda >0$; if $0<N<N^{\ast }$, then there exists a constant ${\lambda }^{\ast }(N)>0$ such that $e_{\lambda }(N)$ admits minimizers for any $\lambda >{\lambda }^{\ast }(N)$, and $e_{\lambda }(N)$ does not admit minimizers for $0<\lambda <{\lambda }^{\ast }(N)$. For any given $0<N<N^{\ast }$, the limit behavior of positive minimizers for $e_{\lambda }(N)$ is also studied as $\lambda \to \infty$, where the mass concentrates at the bottom of $g(x)$.

25 pages, 1 figure