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November  2013, 12(6): 2497-2514. doi: 10.3934/cpaa.2013.12.2497

Super polyharmonic property of solutions for PDE systems and its applications

1. 

College of Mathematics and Information Science, Henan Normal University

2. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

Received  May 2012 Revised  December 2012 Published  May 2013

In this paper, we prove that all the positive solutions for the PDE system \begin{eqnarray} (- \Delta)^k u_i = f_i(u_1, \cdots, u_m), \ x \in R^n, \ i = 1, 2, \cdots, m \ \ \ \ \ (1) \end{eqnarray} are super polyharmonic, i.e. \begin{eqnarray} (- \Delta)^j u_i > 0, \ j=1, 2, \cdots, k-1; \ i =1, 2, \cdots, m. \end{eqnarray}
To prove this important super polyharmonic property, we introduced a few new ideas and derived some new estimates.

As an interesting application, we establish the equivalence between the integral system \begin{eqnarray} u_i(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} f_i(u_1(y), \cdots, u_m(y)) d y, \ x \in R^n \ \ \ \ \ (2) \end{eqnarray} and PDE system (1) when $\alpha = 2k < n.$

In the last few years, a series of results on qualitative properties for solutions of integral systems (2) have been obtained, since the introduction of a powerful tool--the method of moving planes in integral forms. Now due to the equivalence established here, all these properties can be applied to the corresponding PDE systems.

We say that systems (1) and (2) are equivalent, if whenever $u$ is a positive solution of (2), then $u$ is also a solution of \begin{eqnarray} (- \Delta)^k u_i = c f_i(u_1, \cdots, u_m), \ x \in R^n, \ i= 1,2, \cdots, m \end{eqnarray} with some constant $c$; and vice versa.
Citation: Wenxiong Chen, Congming Li. Super polyharmonic property of solutions for PDE systems and its applications. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2497-2514. doi: 10.3934/cpaa.2013.12.2497
References:
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D. Applebaum, "Lévy Processes and Stochastic Calculus,", Second edition, 116 (2009).   Google Scholar

[2]

J. Bertoin, "Lévy Processes,", Cambridge Tracts in Mathematics, 121 (1996).   Google Scholar

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J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media,, Statistical mechanics, 195 (1990).   Google Scholar

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W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm. Pure Appl. Anal., 4 (2005), 1.   Google Scholar

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W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955.   Google Scholar

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W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 4 (2009), 949.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

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W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167.   Google Scholar

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

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W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Disc. Cont. Dyn. Sys., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar

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W. Chen and C. Li, Methods on nonlinear elliptic equations,, AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010).   Google Scholar

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W. Chen and C. Li, A sup + inf inequality near R = 0,, Advances in Math, 220 (2009), 219.  doi: 10.1016/j.aim.2008.09.005.  Google Scholar

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W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., LLVIII (2005), 1.   Google Scholar

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W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347.   Google Scholar

[16]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDEs., 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

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P. Constantin, Euler equations, Navier-Stokes equations and turbulence,, In, 1871 (2006).   Google Scholar

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W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet problems,, J. Math. Anal. Appl., 377 (2011), 744.  doi: 10.1016/j.jmaa.2010.11.035.  Google Scholar

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Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[21]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373.   Google Scholar

[22]

X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potentials,, Comm. Pure. Appl. Anal., 10 (2011), 1111.  doi: 10.3934/cpaa.2011.10.1111.  Google Scholar

[23]

F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. H. Poincare Nonl. Anal., 26 (2009), 1.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[24]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661.   Google Scholar

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C. Jin and C. Li, Quantitative analysis of some system of integral equations,, Cal. Var. & PDEs, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[26]

J. Liu, Y. Guo and Y. Zhang, Liouville-type Theorem for polyharmoic systems in $R^n$,, J. Diff. Equa., 225 (2006), 685.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar

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

[28]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation,, Stud. Appl. Math, 57 (1977), 93.   Google Scholar

[29]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonl. Anal: Theory, 71 (2009), 1796.   Google Scholar

[30]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv., 73 (1998), 206.  doi: 10.1007/s000140050052.  Google Scholar

[31]

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

[32]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar

[33]

Y. Lei, C. Li and Chao Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, Cal. Var. & PDE, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[34]

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

[35]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 6 (2009), 1925.  doi: 10.3934/cpaa.2009.8.1925.  Google Scholar

[36]

D. Li, G. Strohmer and L. Wang, Symmetry of integral equations on bounded domains,, Proc. AMS, 137 (2009), 3695.  doi: 10.1090/S0002-9939-09-09987-0.  Google Scholar

[37]

D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779.   Google Scholar

[38]

A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence,, J. Nonl. Sci., 7 (1997), 9.  doi: 10.1007/BF02679124.  Google Scholar

[39]

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

[40]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 2 (2008), 943.  doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar

[41]

L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system,, Math. and Computer Modelling, 49 (2009), 379.  doi: 10.1016/j.mcm.2008.06.010.  Google Scholar

[42]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adances in Math., 3 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[43]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rat. Mech. Anal., 2 (2010), 455.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[44]

J. Qing and D. Raske, On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds,, International Mathematics Research Notices, (2006), 1.   Google Scholar

[45]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction,, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885.  doi: 10.1016/j.cnsns.2006.03.005.  Google Scholar

[46]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[47]

X. Yan, Liouville-type theorem for a higher order elliptic system,, JMAA, 387 (2012), 153.   Google Scholar

show all references

References:
[1]

D. Applebaum, "Lévy Processes and Stochastic Calculus,", Second edition, 116 (2009).   Google Scholar

[2]

J. Bertoin, "Lévy Processes,", Cambridge Tracts in Mathematics, 121 (1996).   Google Scholar

[3]

J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media,, Statistical mechanics, 195 (1990).   Google Scholar

[4]

L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. Math., 171 (2010), 1903.  doi: 10.2307/2152750.  Google Scholar

[5]

W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations,, Disc. Cont. Dyn. Sys., (2005), 164.   Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm. Pure Appl. Anal., 4 (2005), 1.   Google Scholar

[7]

W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955.   Google Scholar

[8]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 4 (2009), 949.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[9]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Disc. Cont. Dyn. Sys., 4 (2009), 1167.   Google Scholar

[10]

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

[11]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Disc. Cont. Dyn. Sys., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[12]

W. Chen and C. Li, Methods on nonlinear elliptic equations,, AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010).   Google Scholar

[13]

W. Chen and C. Li, A sup + inf inequality near R = 0,, Advances in Math, 220 (2009), 219.  doi: 10.1016/j.aim.2008.09.005.  Google Scholar

[14]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., LLVIII (2005), 1.   Google Scholar

[15]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. Cont. Dyn. Sys., 12 (2005), 347.   Google Scholar

[16]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. PDEs., 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[17]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence,, In, 1871 (2006).   Google Scholar

[18]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Chapman & Hall/CRC Financial Mathematics Series, (2004).   Google Scholar

[19]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet problems,, J. Math. Anal. Appl., 377 (2011), 744.  doi: 10.1016/j.jmaa.2010.11.035.  Google Scholar

[20]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, Adv. Math., 229 (2012), 2835.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[21]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373.   Google Scholar

[22]

X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potentials,, Comm. Pure. Appl. Anal., 10 (2011), 1111.  doi: 10.3934/cpaa.2011.10.1111.  Google Scholar

[23]

F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. H. Poincare Nonl. Anal., 26 (2009), 1.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[24]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. AMS, 134 (2006), 1661.   Google Scholar

[25]

C. Jin and C. Li, Quantitative analysis of some system of integral equations,, Cal. Var. & PDEs, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[26]

J. Liu, Y. Guo and Y. Zhang, Liouville-type Theorem for polyharmoic systems in $R^n$,, J. Diff. Equa., 225 (2006), 685.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar

[27]

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

[28]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation,, Stud. Appl. Math, 57 (1977), 93.   Google Scholar

[29]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, Nonl. Anal: Theory, 71 (2009), 1796.   Google Scholar

[30]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$,, Comment. Math. Helv., 73 (1998), 206.  doi: 10.1007/s000140050052.  Google Scholar

[31]

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

[32]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Comm. Pure Appl. Anal., 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar

[33]

Y. Lei, C. Li and Chao Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, Cal. Var. & PDE, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[34]

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

[35]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, Comm. Pure Appl. Anal., 6 (2009), 1925.  doi: 10.3934/cpaa.2009.8.1925.  Google Scholar

[36]

D. Li, G. Strohmer and L. Wang, Symmetry of integral equations on bounded domains,, Proc. AMS, 137 (2009), 3695.  doi: 10.1090/S0002-9939-09-09987-0.  Google Scholar

[37]

D. Li and R. Zhuo, An integral equation on half space,, Proc. AMS, 138 (2010), 2779.   Google Scholar

[38]

A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence,, J. Nonl. Sci., 7 (1997), 9.  doi: 10.1007/BF02679124.  Google Scholar

[39]

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

[40]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 2 (2008), 943.  doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar

[41]

L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system,, Math. and Computer Modelling, 49 (2009), 379.  doi: 10.1016/j.mcm.2008.06.010.  Google Scholar

[42]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adances in Math., 3 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[43]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rat. Mech. Anal., 2 (2010), 455.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[44]

J. Qing and D. Raske, On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds,, International Mathematics Research Notices, (2006), 1.   Google Scholar

[45]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction,, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885.  doi: 10.1016/j.cnsns.2006.03.005.  Google Scholar

[46]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

[47]

X. Yan, Liouville-type theorem for a higher order elliptic system,, JMAA, 387 (2012), 153.   Google Scholar

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