American Institute of Mathematical Sciences

March  2015, 35(3): 1039-1057. doi: 10.3934/dcds.2015.35.1039

On the integral systems with negative exponents

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

Received  April 2014 Revised  August 2014 Published  October 2014

This paper is concerned with the integral system $$\left \{ \begin{array}{ll} &u(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{v^q(y)},\quad u>0~in~R^n,\\ &v(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{u^p(y)},\quad v>0~in~R^n, \end{array} \right.$$ where $n \geq 1$, $p,q,\lambda \neq 0$. Such an integral system appears in the study of the conformal geometry. We obtain several necessary conditions for the existence of the $C^1$ positive entire solutions, particularly including the critical condition $$\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n},$$ which is the necessary and sufficient condition for the invariant of the system and some energy functionals under the scaling transformation. The necessary condition $\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}$ can be relaxed to another weaker one $\min\{p,q\}>\frac{n+\lambda}{\lambda}$ for the system with double bounded coefficients. In addition, we classify the radial solutions in the case of $p=q$ as the form $$u(x)=v(x)=a(b^2+|x-x_0|^2)^{\frac{\lambda}{2}}$$ with $a,b>0$ and $x_0 \in R^n$. Finally, we also deduce some analogous necessary conditions of existence for the weighted system.
Citation: Yutian Lei. On the integral systems with negative exponents. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1039-1057. doi: 10.3934/dcds.2015.35.1039
References:
 [1] G. Caristi, L. 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. [2] A. Chang and M. del Mar Gonzalez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016. [3] 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. [4] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1. [5] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [6] W. Chen, C. 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. [7] 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. [8] Z. Cheng and C. Li, An extended discrete Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 34 (2014), 1951-1959. doi: 10.3934/dcds.2014.34.1951. [9] Y. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234. doi: 10.1016/j.jde.2008.06.027. [10] J. Davila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Ann., 348 (2010), 143-193. doi: 10.1007/s00208-009-0476-8. [11] 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. [12] Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. doi: 10.3934/dcds.2014.34.2561. [13] Y. Hua and X. Yu, Necessary conditions for existence results of some integral system, Abstr. Appl. Anal., (2013), Art. ID 504282, 5 pp. [14] 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. [15] Y. Lei and C. Li, Sharp Criteria of Liouville Type for some Nonlinear Systems, arXiv:1301.6235, 2013. [16] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. [17] Y. Lei and Z. Lü, Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 33 (2013), 1987-2005. doi: 10.3934/dcds.2013.33.1987. [18] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [19] 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. [20] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [21] L. Ma and J. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017. [22] P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations, (2003), 1-13. [23] Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014. [24] 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. [25] X. Xu, Exact solution of nonlinear conformally invarient integral equations in $R^3$, Adv. Math., 194 (2005), 485-503. doi: 10.1016/j.aim.2004.07.004. [26] X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005. [27] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.

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References:
 [1] G. Caristi, L. 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. [2] A. Chang and M. del Mar Gonzalez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016. [3] 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. [4] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1. [5] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [6] W. Chen, C. 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. [7] 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. [8] Z. Cheng and C. Li, An extended discrete Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 34 (2014), 1951-1959. doi: 10.3934/dcds.2014.34.1951. [9] Y. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234. doi: 10.1016/j.jde.2008.06.027. [10] J. Davila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Ann., 348 (2010), 143-193. doi: 10.1007/s00208-009-0476-8. [11] 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. [12] Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. doi: 10.3934/dcds.2014.34.2561. [13] Y. Hua and X. Yu, Necessary conditions for existence results of some integral system, Abstr. Appl. Anal., (2013), Art. ID 504282, 5 pp. [14] 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. [15] Y. Lei and C. Li, Sharp Criteria of Liouville Type for some Nonlinear Systems, arXiv:1301.6235, 2013. [16] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. [17] Y. Lei and Z. Lü, Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 33 (2013), 1987-2005. doi: 10.3934/dcds.2013.33.1987. [18] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [19] 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. [20] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [21] L. Ma and J. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017. [22] P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations, (2003), 1-13. [23] Ph. Souplet, The proof of the Lane-Emden conjecture in 4 space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014. [24] 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. [25] X. Xu, Exact solution of nonlinear conformally invarient integral equations in $R^3$, Adv. Math., 194 (2005), 485-503. doi: 10.1016/j.aim.2004.07.004. [26] X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005. [27] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.
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