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

Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity

Jianqing Chen 1,

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 $i\varphi_t = - \triangle \varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.
Keywords: Variant Gagliardo-Nirenberg interpolation inequality, sharp variational characterization, minimal action solution, Hartree type nonlinearity..
Mathematics Subject Classification: Primary: 35J20, 35A3.
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:
[1]

P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827.  Google Scholar

[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.  Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, Vol. 10, Providence, Rhode Island, 2005. doi: 10.1090/cln/010.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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E. P. Gross, Physics of many-particle systems, (eds. E. Meeron), New York-London-Paris, Gordon Breash, 1 (1966), 231-406. Google Scholar

[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.  Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. II, IV, Elsevier (Singapore) Pte Ltd, 2003. Google Scholar

[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.  Google Scholar

[14]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  Google Scholar

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M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statistical Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, Vol. 10, Providence, Rhode Island, 2005. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. P. Gross, Physics of many-particle systems, (eds. E. Meeron), New York-London-Paris, Gordon Breash, 1 (1966), 231-406. Google Scholar

[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.  Google Scholar

[12]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. II, IV, Elsevier (Singapore) Pte Ltd, 2003. Google Scholar

[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.  Google Scholar

[14]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.  Google Scholar

[15]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[16]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statistical Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.  Google Scholar

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