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Non-existence of positive solutions for a higher order fractional equation
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians
1. | School of Mathematics and Systems Science, Beihang University (BUAA), Beijing 100083, China |
2. | School of Mathematics, Hunan University, Changsha 410082, China |
3. | School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, NSW, Australia |
In this paper, we are concerned with the fractional order equations (1) with Hartree type $ \dot{H}^{\frac{α}{2}} $-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions $ u $ to (1) and (3) are radially symmetric about some point $ x_{0}∈\mathbb{R}^{d} $ and derive the explicit forms for $ u $ (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).
References:
[1] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[2] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[3] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[4] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[5] |
D. Cao and W. Dai,
Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Royal Soc. Edinburgh-A: Math., 97 (2018), 255-273.
doi: 10.1080/00036811.2016.1260708. |
[6] |
S. A. Chang and P. C. Yang,
On uniqueness of solutions of $ n $-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.
doi: 10.4310/MRL.1997.v4.n1.a9. |
[7] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[8] |
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. |
[9] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. |
[10] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[11] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
W. Chen, C. Li and B. Ou,
Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[13] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, 1-43, Lecture Notes in Math., 1871, Springer, Berlin, 2006. |
[14] |
W. Dai and Z. Liu,
Classification of positive solutions to a system of Hardy-Sobolev type equations, Acta Mathematica Scientia, 37 (2017), 1415-1436.
doi: 10.1016/S0252-9602(17)30082-6. |
[15] |
W. Dai, Z. Liu and G. Lu,
Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, 46 (2017), 569-588.
doi: 10.1007/s11118-016-9594-6. |
[16] |
W. Dai, Z. Liu and G. Lu,
Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space, Comm. Pure Appl. Anal., 16 (2017), 1253-1264.
doi: 10.3934/cpaa.2017061. |
[17] |
Y. Fang and W. Chen,
A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[18] |
J. Frohlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in: Sminaire E. D. P. (2003-2004), Expos nXVIII, (2004), 26pp. |
[19] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbb{R}^{n} $, Mathematical Analysis and Applications, Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[20] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry and related properties via maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[21] |
Y. Lei,
Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.
doi: 10.1137/120879282. |
[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] |
C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[24] |
D. Li, C. Miao and X. Zhang,
The focusing energy-critical Hartree equation, J. Diff. Equations, 246 (2009), 1139-1163.
doi: 10.1016/j.jde.2008.05.013. |
[25] |
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. |
[26] |
E. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
doi: 10.1007/BF01609845. |
[27] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[28] |
C. S. Lin,
A classification of solutions of a conformally invariant fourth order equation in $ \mathbb{R}^{n} $, Comment. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[29] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, parts1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire., 1 (1984), 109-145 and 223--283.
doi: 10.1016/S0294-1449(16)30422-X. |
[30] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case, parts1 and 2, Revista Math. Iberoamericana, 1 (1985), 145-201 and 45--121.
|
[31] |
Z. Liu and W. Dai,
A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.
doi: 10.1515/ans-2015-0106. |
[32] |
S. Liu,
Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.
doi: 10.1016/j.na.2009.01.014. |
[33] |
C. Ma, W. Chen and C. Li,
Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[34] |
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. |
[35] |
C. Miao, G. Xu and L. Zhao,
Global wellposedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Equations, 246 (2009), 3715-3749.
doi: 10.1016/j.jde.2008.11.011. |
[36] |
C. Miao, G. Xu and L. Zhao,
Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.
doi: 10.4064/cm114-2-5. |
[37] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[38] |
B. Ou,
A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184.
|
[39] |
J. Serrin,
A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
|
[40] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970. |
[41] |
J. Wei and X. Xu,
Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[42] |
D. Xu and Y. Lei,
Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.
doi: 10.1016/j.aml.2014.12.007. |
show all references
References:
[1] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[2] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[3] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[4] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[5] |
D. Cao and W. Dai,
Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Royal Soc. Edinburgh-A: Math., 97 (2018), 255-273.
doi: 10.1080/00036811.2016.1260708. |
[6] |
S. A. Chang and P. C. Yang,
On uniqueness of solutions of $ n $-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.
doi: 10.4310/MRL.1997.v4.n1.a9. |
[7] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[8] |
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. |
[9] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. |
[10] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[11] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
W. Chen, C. Li and B. Ou,
Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445. |
[13] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, 1-43, Lecture Notes in Math., 1871, Springer, Berlin, 2006. |
[14] |
W. Dai and Z. Liu,
Classification of positive solutions to a system of Hardy-Sobolev type equations, Acta Mathematica Scientia, 37 (2017), 1415-1436.
doi: 10.1016/S0252-9602(17)30082-6. |
[15] |
W. Dai, Z. Liu and G. Lu,
Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, 46 (2017), 569-588.
doi: 10.1007/s11118-016-9594-6. |
[16] |
W. Dai, Z. Liu and G. Lu,
Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space, Comm. Pure Appl. Anal., 16 (2017), 1253-1264.
doi: 10.3934/cpaa.2017061. |
[17] |
Y. Fang and W. Chen,
A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
doi: 10.1016/j.aim.2012.01.018. |
[18] |
J. Frohlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in: Sminaire E. D. P. (2003-2004), Expos nXVIII, (2004), 26pp. |
[19] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbb{R}^{n} $, Mathematical Analysis and Applications, Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[20] |
B. Gidas, W. Ni and L. Nirenberg,
Symmetry and related properties via maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[21] |
Y. Lei,
Qualitative analysis for the static Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406.
doi: 10.1137/120879282. |
[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] |
C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
doi: 10.1007/s002220050023. |
[24] |
D. Li, C. Miao and X. Zhang,
The focusing energy-critical Hartree equation, J. Diff. Equations, 246 (2009), 1139-1163.
doi: 10.1016/j.jde.2008.05.013. |
[25] |
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. |
[26] |
E. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
doi: 10.1007/BF01609845. |
[27] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[28] |
C. S. Lin,
A classification of solutions of a conformally invariant fourth order equation in $ \mathbb{R}^{n} $, Comment. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[29] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, parts1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire., 1 (1984), 109-145 and 223--283.
doi: 10.1016/S0294-1449(16)30422-X. |
[30] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case, parts1 and 2, Revista Math. Iberoamericana, 1 (1985), 145-201 and 45--121.
|
[31] |
Z. Liu and W. Dai,
A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.
doi: 10.1515/ans-2015-0106. |
[32] |
S. Liu,
Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.
doi: 10.1016/j.na.2009.01.014. |
[33] |
C. Ma, W. Chen and C. Li,
Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[34] |
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. |
[35] |
C. Miao, G. Xu and L. Zhao,
Global wellposedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Diff. Equations, 246 (2009), 3715-3749.
doi: 10.1016/j.jde.2008.11.011. |
[36] |
C. Miao, G. Xu and L. Zhao,
Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.
doi: 10.4064/cm114-2-5. |
[37] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[38] |
B. Ou,
A Remark on a singular integral equation, Houston J. Math., 25 (1999), 181-184.
|
[39] |
J. Serrin,
A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
|
[40] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970. |
[41] |
J. Wei and X. Xu,
Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.
doi: 10.1007/s002080050258. |
[42] |
D. Xu and Y. Lei,
Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.
doi: 10.1016/j.aml.2014.12.007. |
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