November  2018, 38(11): 5351-5377. doi: 10.3934/dcds.2018236

Liouville theorems and classification results for a nonlocal Schrödinger equation

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received  March 2017 Revised  June 2017 Published  August 2018

Fund Project: The research was supported by NSF of China (No. 11471164, 11671209), and PAPD of Jiangsu Higher Education Institutions

In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation
$-Δ u = pu^{p-1}(|x|^{2-n}*u^p),\;\; u>0 \;\;in\;\; R^n, $
where
$n ≥ 3$
and
$p≥ 1$
. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when
$1 ≤ p <\frac{n+2}{n-2}$
by means of the method of moving planes to the following system
$\left\{ \begin{array}{l} - \Delta u = \sqrt p {u^{p - 1}}v,\;\;u > 0\;\;in\;\;{R^n},\\ - \Delta v = \sqrt p {u^p},\;\;v > 0\;\;in\;\;{R^n}.\end{array} \right.$
When
$p = \frac{n+2}{n-2}$
, all the positive solutions can be classified as
$u(x) = c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$
with the help of an integral system involving the Newton potential, where
$c, t$
are positive constants, and
$x^* ∈ R^n$
. In addition, we also give other equivalent conditions to classify those positive solutions. When
$p>\frac{n+2}{n-2}$
, by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate
$\frac{2}{p-1}$
. Finally, we point out that the equation has positive stable solutions if and only if
$p ≥ 1+\frac{4}{n-4-2\sqrt{n-1}}$
.
Citation: Yutian Lei. Liouville theorems and classification results for a nonlocal Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5351-5377. doi: 10.3934/dcds.2018236
References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[2]

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W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010.  Google Scholar

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W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

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W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

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A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $R^N$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

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F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, 2003,327–335.  Google Scholar

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B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981,369–402.  Google Scholar

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[14]

J. Ginibre and G. Velo, On a class of non linear Schrödinger equations with non local interaction, Math. Z., 170 (1980), 109-136.  doi: 10.1007/BF01214768.  Google Scholar

[15]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations, Ⅱ, Annales Henri Poincare, 1 (2000), 753-800.  doi: 10.1007/PL00001014.  Google Scholar

[16]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[17]

Z. Guo and J. Wei, Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc., 363 (2011), 4777-4799.  doi: 10.1090/S0002-9947-2011-05292-X.  Google Scholar

[18]

E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54.  doi: 10.1007/s00526-012-0509-0.  Google Scholar

[19]

L. Jeanjean and T. Luo, Sharp nonexistence results of prescribed $L^2$-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.  doi: 10.1007/s00033-012-0272-2.  Google Scholar

[20]

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[21]

D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Meth. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[22]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6.  Google Scholar

[23]

Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282.  Google Scholar

[24]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[25]

C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.  doi: 10.1137/080712301.  Google Scholar

[26]

D. LiC. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations, 246 (2009), 1139-1163.  doi: 10.1016/j.jde.2008.05.013.  Google Scholar

[27]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  doi: 10.4171/JEMS/6.  Google Scholar

[28]

Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.  Google Scholar

[29]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[30]

Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$, Duke Math. J., 57 (1988), 895-924.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

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E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[32]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[33]

E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845.  Google Scholar

[34]

B. Liu and L. Ma, Invariant sets and the blow up threshold for a nonlocal equation of parabolic type, Nonlinear Anal., 110 (2014), 141-156.  doi: 10.1016/j.na.2014.08.004.  Google Scholar

[35]

J. LiuY. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.   Google Scholar

[36]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[37]

L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole Space, Adv. Math., 225 (2010), 3052-3063.  doi: 10.1016/j.aim.2010.05.022.  Google Scholar

[38]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[39]

V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.  Google Scholar

[40]

K. Nakanishi, Energy scattering for Hartree equations, Math. Res. Lett., 6 (1999), 107-118.  doi: 10.4310/MRL.1999.v6.n1.a8.  Google Scholar

[41]

P. Quittner and Ph. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.  Google Scholar

[42]

S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882.  doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[43]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[2]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. Dyn. Sys., Vol. 4, 2010.  Google Scholar

[7]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[8]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[9]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar

[10]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $R^N$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[11]

F. Gazzola, Critical exponents which relate embedding inequalities with quasilinear elliptic operator, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, 2003,327–335.  Google Scholar

[12]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981,369–402.  Google Scholar

[13]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[14]

J. Ginibre and G. Velo, On a class of non linear Schrödinger equations with non local interaction, Math. Z., 170 (1980), 109-136.  doi: 10.1007/BF01214768.  Google Scholar

[15]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations, Ⅱ, Annales Henri Poincare, 1 (2000), 753-800.  doi: 10.1007/PL00001014.  Google Scholar

[16]

C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[17]

Z. Guo and J. Wei, Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc., 363 (2011), 4777-4799.  doi: 10.1090/S0002-9947-2011-05292-X.  Google Scholar

[18]

E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54.  doi: 10.1007/s00526-012-0509-0.  Google Scholar

[19]

L. Jeanjean and T. Luo, Sharp nonexistence results of prescribed $L^2$-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.  doi: 10.1007/s00033-012-0272-2.  Google Scholar

[20]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[21]

D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Meth. Anal., 49 (1972/73), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[22]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6.  Google Scholar

[23]

Y. Lei, Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282.  Google Scholar

[24]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[25]

C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. Math. Anal., 40 (2008), 1049-1057.  doi: 10.1137/080712301.  Google Scholar

[26]

D. LiC. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations, 246 (2009), 1139-1163.  doi: 10.1016/j.jde.2008.05.013.  Google Scholar

[27]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  doi: 10.4171/JEMS/6.  Google Scholar

[28]

Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.  Google Scholar

[29]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[30]

Y. Li and W.-M. Ni, On conformal scalar curvature equations in $R^n$, Duke Math. J., 57 (1988), 895-924.  doi: 10.1215/S0012-7094-88-05740-7.  Google Scholar

[31]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[32]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[33]

E. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845.  Google Scholar

[34]

B. Liu and L. Ma, Invariant sets and the blow up threshold for a nonlocal equation of parabolic type, Nonlinear Anal., 110 (2014), 141-156.  doi: 10.1016/j.na.2014.08.004.  Google Scholar

[35]

J. LiuY. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.   Google Scholar

[36]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[37]

L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole Space, Adv. Math., 225 (2010), 3052-3063.  doi: 10.1016/j.aim.2010.05.022.  Google Scholar

[38]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[39]

V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.  Google Scholar

[40]

K. Nakanishi, Energy scattering for Hartree equations, Math. Res. Lett., 6 (1999), 107-118.  doi: 10.4310/MRL.1999.v6.n1.a8.  Google Scholar

[41]

P. Quittner and Ph. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.  Google Scholar

[42]

S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal., 263 (2012), 3857-3882.  doi: 10.1016/j.jfa.2012.09.012.  Google Scholar

[43]

X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar

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