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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[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.

[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.

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