Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity

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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

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

Received June 2015 Revised August 2016 Published November 2016

Abstract
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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.
[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. doi: 10.1088/0951-7715/21/5/001.
[3]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (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. 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. 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. 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. 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. 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. doi: 10.1063/1.523491.
[10]	E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231.
[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.
[12]	M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (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. doi: 10.1137/S0036141003431530.
[14]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567.
[15]	M. Willem, Minimax Theorems,, Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1.
[16]	J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731. doi: 10.1023/A:1026437923987.

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.
[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. doi: 10.1088/0951-7715/21/5/001.
[3]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (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. 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. 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. 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. 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. 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. doi: 10.1063/1.523491.
[10]	E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231.
[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.
[12]	M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (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. doi: 10.1137/S0036141003431530.
[14]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567.
[15]	M. Willem, Minimax Theorems,, Birkhäuser, (1996). doi: 10.1007/978-1-4612-4146-1.
[16]	J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731. doi: 10.1023/A:1026437923987.

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