# American Institute of Mathematical Sciences

December  2016, 9(6): 1613-1628. doi: 10.3934/dcdss.2016066

## Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity

 1 School of Mathematics and Computer Science, Fujian Normal University, Qishan Campus, Fuzhou 350117, China

Received  June 2015 Revised  August 2016 Published  November 2016

In this paper, we first give a sharp variational characterization to the smallest positive constant $C_{VGN}$ in the following Variant Gagliardo-Nirenberg interpolation inequality: $$\int_{\mathbb{R}^N\times\mathbb{R}^N}{{|u(x)|^p|u(y)|^p}\over{|x-y|^\alpha}}dxdy\leq C_{VGN} \|\nabla u\|_{L^2}^{N(p-2)+\alpha} \|u\|_{L^2}^{2p-(N(p-2)+\alpha)},$$ where $u\in W^{1,2}(\mathbb{R}^N)$ and $N\geq 1$. Then we use this characterization to determine the sharp threshold of $\|\varphi_0\|_{L^2}$ such that the solution of $i\varphi_t = - \triangle \varphi + |x|^2\varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$ with initial condition $\varphi(0, x) = \varphi_0$ exists globally or blows up in a finite time. We also outline some results on the applications of $C_{VGN}$ to the Cauchy problem of $i\varphi_t = - \triangle \varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.
Citation: Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
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