# 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 and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
##### References:
 [1] P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. [2] Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 879-898. doi: 10.1088/0951-7715/21/5/001. [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, Vol. 10, Providence, Rhode Island, 2005. doi: 10.1090/cln/010. [4] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504. [5] G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598. doi: 10.1016/j.jmaa.2005.07.008. [6] J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D, 227 (2007), 142-148. doi: 10.1016/j.physd.2007.01.004. [7] J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential, Commun. Math. Sci., 7 (2009), 549-570. doi: 10.4310/CMS.2009.v7.n3.a2. [8] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem, J. Funct. Anal., 32 (1979), 33-71. doi: 10.1016/0022-1236(79)90077-6. [9] R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491. [10] E. P. Gross, Physics of many-particle systems, (eds. E. Meeron), New York-London-Paris, Gordon Breash, 1 (1966), 231-406. [11] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145 and 223-283. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. II, IV, Elsevier (Singapore) Pte Ltd, 2003. [13] M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons, SIAM J. Math. Anal., 36 (2004), 967-985. doi: 10.1137/S0036141003431530. [14] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. [15] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [16] J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statistical Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.

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##### References:
 [1] P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. [2] Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 879-898. doi: 10.1088/0951-7715/21/5/001. [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, Vol. 10, Providence, Rhode Island, 2005. doi: 10.1090/cln/010. [4] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504. [5] G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598. doi: 10.1016/j.jmaa.2005.07.008. [6] J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D, 227 (2007), 142-148. doi: 10.1016/j.physd.2007.01.004. [7] J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential, Commun. Math. Sci., 7 (2009), 549-570. doi: 10.4310/CMS.2009.v7.n3.a2. [8] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem, J. Funct. Anal., 32 (1979), 33-71. doi: 10.1016/0022-1236(79)90077-6. [9] R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491. [10] E. P. Gross, Physics of many-particle systems, (eds. E. Meeron), New York-London-Paris, Gordon Breash, 1 (1966), 231-406. [11] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145 and 223-283. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. II, IV, Elsevier (Singapore) Pte Ltd, 2003. [13] M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons, SIAM J. Math. Anal., 36 (2004), 967-985. doi: 10.1137/S0036141003431530. [14] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. [15] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [16] J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statistical Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.
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