# American Institute of Mathematical Sciences

November  2013, 12(6): 2601-2613. doi: 10.3934/cpaa.2013.12.2601

## Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications

 1 Department of Applied Mathematics, Key Laboratory of Space Applied Physics and Chemistry, Ministry of Education, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China, China

Received  July 2012 Revised  January 2013 Published  May 2013

Let $R^n_+$ be the $n$-dimensional upper half Euclidean space, $m$ be a positive integer. In this paper, we consider the following system of integral equations on $R^n_+$: \begin{eqnarray} u(x)=\int_{R^n_+}G(x,y)v^q(y)dy, \\ v(x)=\int_{R^n_+}G(x,y)u^p(y)dy, \end{eqnarray} where \begin{eqnarray} G(x,y)=\frac{c_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_ny_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{\frac{n}{2}}}dz \end{eqnarray} with $0 < 2m < n$ and $p,q>1$. Nonexistence of positive solution is proved by using the method of moving planes in integral forms. We also obtain the equivalence between the system of integral equations and corresponding partial differential equations.
Citation: Dongyan Li, Yongzhong Wang. Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2601-2613. doi: 10.3934/cpaa.2013.12.2601
##### References:
 [1] W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirchlet problems,, Math.Z., 261 (2009), 805.  doi: 10.1007/s00209-008-0352-3.  Google Scholar [2] 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 [3] Y. Fang and J. Zhang, Nonexistence of positive solition for an integral equation on $R^n_+$,, Commun. Pure Appl. Anal., 12 (2013), 663.  doi: 10.3934/cpaa.2013.12.663.  Google Scholar [4] C. S. 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 [5] J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar [6] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math, 59 (2006), 330.  doi: 10.1002/cpa.20130.  Google Scholar [7] W. Chen, C. Li and B. Ou, Qualitative problems of solutions for a system of integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.   Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, appear to in Commun. Pure Appl. Anal., (2012).   Google Scholar [10] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1.  doi: 10.3934/cpaa.2005.4.1.  Google Scholar [11] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 4 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [12] C. Jin and C. Li, Symmetry of solutions to some system of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [13] W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Differ. Equ. Dyn. Syst., 4 (2010).  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [14] C. Jin and C. Li, Quantitative analysis of some system of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar [15] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar [16] W. Chen, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst., 12 (2005), 347.   Google Scholar [17] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations,, Commun. Pure Appl. Anal., 6 (2009), 1925.  doi: 10.3934/cpaa.2009.8.1925.  Google Scholar [18] L. Ma and D. Z. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar [19] L. Ma and D. Z. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943.  doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar [20] B. Ou, A Remark on a singular integral equation,, Houston J. Math., 25 (1999), 181.   Google Scholar [21] J. Liu, Y. Guo and Y. Zhang, Liouville type theorems for polyharmonic system in $R^n$,, J. Differential Equations, 225 (2006), 685.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar

show all references

##### References:
 [1] W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirchlet problems,, Math.Z., 261 (2009), 805.  doi: 10.1007/s00209-008-0352-3.  Google Scholar [2] 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 [3] Y. Fang and J. Zhang, Nonexistence of positive solition for an integral equation on $R^n_+$,, Commun. Pure Appl. Anal., 12 (2013), 663.  doi: 10.3934/cpaa.2013.12.663.  Google Scholar [4] C. S. 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 [5] J. Wei and X. Xu, Classification of solution of higer order conformally invariant equations,, Math. Ann., 313 (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar [6] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math, 59 (2006), 330.  doi: 10.1002/cpa.20130.  Google Scholar [7] W. Chen, C. Li and B. Ou, Qualitative problems of solutions for a system of integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.   Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, appear to in Commun. Pure Appl. Anal., (2012).   Google Scholar [10] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1.  doi: 10.3934/cpaa.2005.4.1.  Google Scholar [11] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 4 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [12] C. Jin and C. Li, Symmetry of solutions to some system of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [13] W. Chen and C. Li, "Methods on Nonlinear Elliptic Equations,", AIMS Book Series on Differ. Equ. Dyn. Syst., 4 (2010).  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [14] C. Jin and C. Li, Quantitative analysis of some system of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar [15] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar [16] W. Chen, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst., 12 (2005), 347.   Google Scholar [17] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integal equations,, Commun. Pure Appl. Anal., 6 (2009), 1925.  doi: 10.3934/cpaa.2009.8.1925.  Google Scholar [18] L. Ma and D. Z. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar [19] L. Ma and D. Z. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943.  doi: 10.1016/j.jmaa.2007.12.064.  Google Scholar [20] B. Ou, A Remark on a singular integral equation,, Houston J. Math., 25 (1999), 181.   Google Scholar [21] J. Liu, Y. Guo and Y. Zhang, Liouville type theorems for polyharmonic system in $R^n$,, J. Differential Equations, 225 (2006), 685.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar
 [1] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [2] Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109 [3] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [4] Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021001 [5] Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278 [6] Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104 [7] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [8] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [9] Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 [10] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [11] Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 [12] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [13] Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101 [14] Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067 [15] Gaojun Luo, Xiwang Cao. Two classes of near-optimal codebooks with respect to the Welch bound. Advances in Mathematics of Communications, 2021, 15 (2) : 279-289. doi: 10.3934/amc.2020066 [16] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [17] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [18] Kengo Matsumoto. $C^*$-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, , () : -. doi: 10.3934/era.2021006 [19] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [20] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

2019 Impact Factor: 1.105