A quasilinear Schrödinger equation with Hartree type nonlinearity

Abstract: In this paper, we deal with the Cauchy problem of the quasilinear Sch\"{o}dinger equation \begin{equation*} \left{ \begin{array}{lll} iu_t=\Delta u+2uh'(|u|^2)\Delta h(|u|^2)+(W(x)\ast|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $W(x)$ are some real valued functions. Our focus is to investigate how the interplay between the potential $W(x)$ and the quasilinear presence $h(s)$ affects the blowup in finite time and global existence of the solution. In a special, we can obtain the watershed condition on $W(x)$ in the following sense: If $W(x)\in L^1(\mathbb{R}^N)\cap {L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)} $, then exist $q_c$ and $q_s$ such that the solution is global existence for any initial data in the energy space when $q>q_c$ and the solution maybe blow up in finite time for some initial data when $q_s<q<q_c$, and for $q=q_c$ whether the solution is global existence or not depend on the initial data.