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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 and Applied Analysis, 2013, 12 (6) : 2497-2514. doi: 10.3934/cpaa.2013.12.2497
##### References:
 [1] D. Applebaum, "Lévy Processes and Stochastic Calculus," Second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. [2] J. Bertoin, "Lévy Processes," Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996. [3] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, Models and Physical Applications, Physics Reports, 195 (1990). [4] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.2307/2152750. [5] W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Disc. Cont. Dyn. Sys., \textbfS (2005), 164-172. [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. [7] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. AMS, 136 (2008), 955-962. [8] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 4 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [9] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184. [10] 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. [11] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [12] W. Chen and C. Li, Methods on nonlinear elliptic equations, AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010). [13] W. Chen and C. Li, A sup + inf inequality near R = 0, Advances in Math, 220 (2009), 219-245. doi: 10.1016/j.aim.2008.09.005. [14] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., LLVIII (2005), 1-14. [15] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [16] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDEs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [17] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, In "Mathematical Foundation of Turbulent Viscous Flows," Vol. 1871 of Lecture Notes in Math. 1?3, Springer, Berlin, 2006. [18] R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Chapman & Hall/CRC Financial Mathematics Series, Boca Raton, Fl, 2004. [19] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753. doi: 10.1016/j.jmaa.2010.11.035. [20] 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. [21] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. [22] X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potentials, Comm. Pure. Appl. Anal., 10 (2011), 1111-1119. doi: 10.3934/cpaa.2011.10.1111. [23] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006. [24] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. AMS, 134 (2006), 1661-1670. [25] C. Jin and C. Li, Quantitative analysis of some system of integral equations, Cal. Var. & PDEs, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. [26] J. Liu, Y. Guo and Y. Zhang, Liouville-type Theorem for polyharmoic systems in $R^n$, J. Diff. Equa., 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. [27] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374. doi: 10.2307/2007032. [28] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math, 57 (1977), 93-105. [29] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonl. Anal: Theory, Methods & Appl., 71 (2009), 1796-1806. [30] C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. [31] Y. Li, Remarks on some conformally invariant integral equations: the method of moving spheres, J. Euro. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6. [32] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [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), 43-61. doi: 10.1007/s00526-011-0450-7. [34] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. of Appl. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [35] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [36] D. Li, G. Strohmer and L. Wang, Symmetry of integral equations on bounded domains, Proc. AMS, 137 (2009), 3695-3702. doi: 10.1090/S0002-9939-09-09987-0. [37] D. Li and R. Zhuo, An integral equation on half space, Proc. AMS, 138 (2010), 2779-2791. [38] A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonl. Sci., 7 (1997), 9-44. doi: 10.1007/BF02679124. [39] 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. [40] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 2 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [41] L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system, Math. and Computer Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [42] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adances in Math., 3 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [43] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rat. Mech. Anal., 2 (2010) 455-467. doi: 10.1007/s00205-008-0208-3. [44] J. Qing and D. Raske, On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds, International Mathematics Research Notices, Vol. 2006, Article ID 94172, 1-20. [45] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889. doi: 10.1016/j.cnsns.2006.03.005. [46] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. [47] X. Yan, Liouville-type theorem for a higher order elliptic system, JMAA, 387 (2012), 153-165.

show all references

##### References:
 [1] D. Applebaum, "Lévy Processes and Stochastic Calculus," Second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. [2] J. Bertoin, "Lévy Processes," Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996. [3] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, Models and Physical Applications, Physics Reports, 195 (1990). [4] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.2307/2152750. [5] W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Disc. Cont. Dyn. Sys., \textbfS (2005), 164-172. [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. [7] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. AMS, 136 (2008), 955-962. [8] W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 4 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [9] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184. [10] 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. [11] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [12] W. Chen and C. Li, Methods on nonlinear elliptic equations, AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010). [13] W. Chen and C. Li, A sup + inf inequality near R = 0, Advances in Math, 220 (2009), 219-245. doi: 10.1016/j.aim.2008.09.005. [14] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., LLVIII (2005), 1-14. [15] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [16] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDEs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [17] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, In "Mathematical Foundation of Turbulent Viscous Flows," Vol. 1871 of Lecture Notes in Math. 1?3, Springer, Berlin, 2006. [18] R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Chapman & Hall/CRC Financial Mathematics Series, Boca Raton, Fl, 2004. [19] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Drichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753. doi: 10.1016/j.jmaa.2010.11.035. [20] 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. [21] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. [22] X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potentials, Comm. Pure. Appl. Anal., 10 (2011), 1111-1119. doi: 10.3934/cpaa.2011.10.1111. [23] F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006. [24] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. AMS, 134 (2006), 1661-1670. [25] C. Jin and C. Li, Quantitative analysis of some system of integral equations, Cal. Var. & PDEs, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. [26] J. Liu, Y. Guo and Y. Zhang, Liouville-type Theorem for polyharmoic systems in $R^n$, J. Diff. Equa., 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016. [27] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374. doi: 10.2307/2007032. [28] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math, 57 (1977), 93-105. [29] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonl. Anal: Theory, Methods & Appl., 71 (2009), 1796-1806. [30] C. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052. [31] Y. Li, Remarks on some conformally invariant integral equations: the method of moving spheres, J. Euro. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6. [32] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [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), 43-61. doi: 10.1007/s00526-011-0450-7. [34] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. of Appl. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [35] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [36] D. Li, G. Strohmer and L. Wang, Symmetry of integral equations on bounded domains, Proc. AMS, 137 (2009), 3695-3702. doi: 10.1090/S0002-9939-09-09987-0. [37] D. Li and R. Zhuo, An integral equation on half space, Proc. AMS, 138 (2010), 2779-2791. [38] A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonl. Sci., 7 (1997), 9-44. doi: 10.1007/BF02679124. [39] 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. [40] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 2 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [41] L. Ma and D. Chen, Radial symmetry and uniqueness of non-negative solutions to an integral system, Math. and Computer Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [42] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adances in Math., 3 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [43] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rat. Mech. Anal., 2 (2010) 455-467. doi: 10.1007/s00205-008-0208-3. [44] J. Qing and D. Raske, On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds, International Mathematics Research Notices, Vol. 2006, Article ID 94172, 1-20. [45] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889. doi: 10.1016/j.cnsns.2006.03.005. [46] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258. [47] X. Yan, Liouville-type theorem for a higher order elliptic system, JMAA, 387 (2012), 153-165.
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