    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
##### References:
  P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,, Adv. Math. Sci. Appl., 12 (2002), 817. Google Scholar  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.  doi: 10.1088/0951-7715/21/5/001.  Google Scholar  T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (2005).  doi: 10.1090/cln/010. Google Scholar  T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549.  doi: 10.1007/BF01403504.  Google Scholar  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.  doi: 10.1016/j.jmaa.2005.07.008.  Google Scholar  J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation,, Phys. D, 227 (2007), 142.  doi: 10.1016/j.physd.2007.01.004.  Google Scholar  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.  doi: 10.4310/CMS.2009.v7.n3.a2.  Google Scholar  J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem,, J. Funct. Anal., 32 (1979), 33.  doi: 10.1016/0022-1236(79)90077-6.  Google Scholar  R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,, J. Math. Phys., 18 (1977), 1794.  doi: 10.1063/1.523491.  Google Scholar  E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231.   Google Scholar  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. Google Scholar  M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (2003).   Google Scholar  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.  doi: 10.1137/S0036141003431530.  Google Scholar  M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567. Google Scholar  M. Willem, Minimax Theorems,, Birkhäuser, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar  J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731.  doi: 10.1023/A:1026437923987.  Google Scholar

show all references

##### References:
  P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,, Adv. Math. Sci. Appl., 12 (2002), 817. Google Scholar  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.  doi: 10.1088/0951-7715/21/5/001.  Google Scholar  T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (2005).  doi: 10.1090/cln/010. Google Scholar  T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549.  doi: 10.1007/BF01403504.  Google Scholar  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.  doi: 10.1016/j.jmaa.2005.07.008.  Google Scholar  J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation,, Phys. D, 227 (2007), 142.  doi: 10.1016/j.physd.2007.01.004.  Google Scholar  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.  doi: 10.4310/CMS.2009.v7.n3.a2.  Google Scholar  J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem,, J. Funct. Anal., 32 (1979), 33.  doi: 10.1016/0022-1236(79)90077-6.  Google Scholar  R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,, J. Math. Phys., 18 (1977), 1794.  doi: 10.1063/1.523491.  Google Scholar  E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231.   Google Scholar  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. Google Scholar  M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (2003).   Google Scholar  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.  doi: 10.1137/S0036141003431530.  Google Scholar  M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567. Google Scholar  M. Willem, Minimax Theorems,, Birkhäuser, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar  J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731.  doi: 10.1023/A:1026437923987.  Google Scholar
  Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011  Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621  Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101  YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1  Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146  C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519  James Nolen, Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1217-1234. doi: 10.3934/dcds.2005.13.1217  Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347  Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437  Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106  Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084  James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137  Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531  Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261  Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081  Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002  S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155  Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206  Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277  Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096

2018 Impact Factor: 0.545