Energy methods for Hartree type equations with inverse-square potentials

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September 2013, 2(3): 531-542. doi: 10.3934/eect.2013.2.531

Energy methods for Hartree type equations with inverse-square potentials

1.	Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received July 2012 Revised February 2013 Published July 2013

Abstract
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Nonlinear Schrödinger equations with nonlocal nonlinearities described by integral operators are considered. This generalizes usual Hartree type equations (HE)$_{0}$. We construct weak solutions to (HE)$_{a}$, $a\neq 0$, even if the kernel is of non-convolution type. The advantage of our methods is the applicability to the problem with strongly singular potential $a|x|^{-2}$ as a term in the linear part and with critical nonlinearity.

Keywords: integral operator, Hartree equations, nonlinear Schrödinger equations, inverse-square potentials., Energy methods.

Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q1.

Citation: 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

References:

[1]	H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,", Grundlehren der Mathematischen Wissenschaften, 343 (2011). doi: 10.1007/978-3-642-16830-7. Google Scholar
[2]	N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential,, J. Funct. Anal., 203 (2003), 519. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar
[3]	N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay,, Indiana Univ. Math. J., 53 (2004), 1665. doi: 10.1512/iumj.2004.53.2541. Google Scholar
[4]	T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar
[5]	T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$,, Manuscripta Math., 61 (1988), 477. doi: 10.1007/BF01258601. Google Scholar
[6]	J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations,, J. Math. Phys., 16 (1975), 1122. doi: 10.1063/1.522642. Google Scholar
[7]	J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction,, Math. Z., 170 (1980), 109. doi: 10.1007/BF01214768. Google Scholar
[8]	E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,, Comm. Math. Phys., 53 (1977), 185. doi: 10.1007/BF01609845. Google Scholar
[9]	N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials,, Appl. Anal., 91 (2012), 1605. doi: 10.1080/00036811.2011.631914. Google Scholar
[10]	N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations,, Evolution Equations and Control Theory, 1 (2012), 337. doi: 10.3934/eect.2012.1.337. Google Scholar
[11]	V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces,, Math. Z., 260 (2008), 377. doi: 10.1007/s00209-007-0279-0. Google Scholar

show all references

References:

[1]	H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,", Grundlehren der Mathematischen Wissenschaften, 343 (2011). doi: 10.1007/978-3-642-16830-7. Google Scholar
[2]	N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential,, J. Funct. Anal., 203 (2003), 519. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar
[3]	N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay,, Indiana Univ. Math. J., 53 (2004), 1665. doi: 10.1512/iumj.2004.53.2541. Google Scholar
[4]	T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003). Google Scholar
[5]	T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$,, Manuscripta Math., 61 (1988), 477. doi: 10.1007/BF01258601. Google Scholar
[6]	J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations,, J. Math. Phys., 16 (1975), 1122. doi: 10.1063/1.522642. Google Scholar
[7]	J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction,, Math. Z., 170 (1980), 109. doi: 10.1007/BF01214768. Google Scholar
[8]	E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,, Comm. Math. Phys., 53 (1977), 185. doi: 10.1007/BF01609845. Google Scholar
[9]	N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials,, Appl. Anal., 91 (2012), 1605. doi: 10.1080/00036811.2011.631914. Google Scholar
[10]	N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations,, Evolution Equations and Control Theory, 1 (2012), 337. doi: 10.3934/eect.2012.1.337. Google Scholar
[11]	V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces,, Math. Z., 260 (2008), 377. doi: 10.1007/s00209-007-0279-0. Google Scholar

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