December  2016, 36(12): 6767-6780. doi: 10.3934/dcds.2016094

Classification of positive solutions to a Lane-Emden type integral system with negative exponents

1. 

School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100

2. 

School of National Fiscal Development, Central University of Finance and Economics, Beijing 100081, China

3. 

School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539, United States

Received  January 2016 Revised  April 2016 Published  October 2016

In this paper, we classify the positive solutions to the following Lane-Emden type integral system with negative exponents \begin{equation*} \begin{cases} u(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau} u^{-p}(y)v^{-q}(y) \, dy, ~x\in \mathbb{R}^{n}, \\ v(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau}u^{-r}(y)v^{-s}(y) \, dy,~ x\in \mathbb{R}^{n}, \end{cases} \end{equation*}where $n \geq 1$ is an integer and $ \tau, p,q,r,s>0.$ Particularly, using an integral form of the method of moving spheres, we classify the positive solutions to the integral system whenever $$p+q=r+s=1 + 2n/\tau.$$ We also establish the non-existence of positive solutions under the condition $$\max\{p+q,r+s\} \leq 1 + 2n/\tau \,\text{ and }\, p + q + r + s < 2(1 + 2n/\tau).$$
Citation: Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094
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.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[2]

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.20116.  Google Scholar

[3]

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

[4]

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

[5]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

J. Dou, Liouville type theorems for the system of integral equations,, Appl. Math. Comput., 217 (2010), 2586.  doi: 10.1016/j.amc.2010.07.071.  Google Scholar

[7]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space,, Int. Math. Res. Not., 3 (2015), 651.   Google Scholar

[8]

J. Dou and M. Zhu, Reversed Hardy-Littlewood-Sobolev inequality,, Int. Math. Res. Not., 19 (2015), 9696.  doi: 10.1093/imrn/rnu241.  Google Scholar

[9]

M. Ghergu, Lane-Emden systems with negative exponents,, J. Funct. Anal., 258 (2010), 3295.  doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[10]

Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications., J. Differential Equations, 260 (2016), 1.  doi: 10.1016/j.jde.2015.06.032.  Google Scholar

[11]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[12]

Y. Lei, On the integral systems with negative exponents,, Discrete Contin. Dyn. Syst., 35 (2015), 1039.  doi: 10.3934/dcds.2015.35.1039.  Google Scholar

[13]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, Discrete Contin. Dyn. Syst., 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[14]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems,, Proc. Amer. Math. Soc., 144 (2016), 3731.  doi: 10.1090/proc/13166.  Google Scholar

[15]

C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, Comm. Partial Differential Equations, 41 (2016), 1029.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

[16]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[17]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27.  doi: 10.1007/BF02786551.  Google Scholar

[18]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc. (JEMS), 6 (2004), 153.   Google Scholar

[19]

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

[20]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, J. Partial Differential Equations, 19 (2006), 256.   Google Scholar

[21]

L. Ma and D. 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

[22]

Q. A. Ngô and V. H. Nguyen, Sharp Reversed Hardy-Littlewood-Sobolev inequality on $\mathbfR^n$,, Israel J. Math., ().   Google Scholar

[23]

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

[24]

X. Xu, Uniqueness theorem for integral equations and its application,, J. Funct. Anal., 247 (2007), 95.  doi: 10.1016/j.jfa.2007.03.005.  Google Scholar

[25]

Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness,, Nonlinear Anal., 74 (2011), 5544.  doi: 10.1016/j.na.2011.05.038.  Google Scholar

show all references

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.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[2]

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.20116.  Google Scholar

[3]

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

[4]

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

[5]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

J. Dou, Liouville type theorems for the system of integral equations,, Appl. Math. Comput., 217 (2010), 2586.  doi: 10.1016/j.amc.2010.07.071.  Google Scholar

[7]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space,, Int. Math. Res. Not., 3 (2015), 651.   Google Scholar

[8]

J. Dou and M. Zhu, Reversed Hardy-Littlewood-Sobolev inequality,, Int. Math. Res. Not., 19 (2015), 9696.  doi: 10.1093/imrn/rnu241.  Google Scholar

[9]

M. Ghergu, Lane-Emden systems with negative exponents,, J. Funct. Anal., 258 (2010), 3295.  doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[10]

Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications., J. Differential Equations, 260 (2016), 1.  doi: 10.1016/j.jde.2015.06.032.  Google Scholar

[11]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[12]

Y. Lei, On the integral systems with negative exponents,, Discrete Contin. Dyn. Syst., 35 (2015), 1039.  doi: 10.3934/dcds.2015.35.1039.  Google Scholar

[13]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, Discrete Contin. Dyn. Syst., 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[14]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems,, Proc. Amer. Math. Soc., 144 (2016), 3731.  doi: 10.1090/proc/13166.  Google Scholar

[15]

C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, Comm. Partial Differential Equations, 41 (2016), 1029.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

[16]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[17]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27.  doi: 10.1007/BF02786551.  Google Scholar

[18]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc. (JEMS), 6 (2004), 153.   Google Scholar

[19]

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

[20]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, J. Partial Differential Equations, 19 (2006), 256.   Google Scholar

[21]

L. Ma and D. 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

[22]

Q. A. Ngô and V. H. Nguyen, Sharp Reversed Hardy-Littlewood-Sobolev inequality on $\mathbfR^n$,, Israel J. Math., ().   Google Scholar

[23]

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

[24]

X. Xu, Uniqueness theorem for integral equations and its application,, J. Funct. Anal., 247 (2007), 95.  doi: 10.1016/j.jfa.2007.03.005.  Google Scholar

[25]

Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness,, Nonlinear Anal., 74 (2011), 5544.  doi: 10.1016/j.na.2011.05.038.  Google Scholar

[1]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[2]

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

[3]

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

[4]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030

[5]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083

[6]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230

[7]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[8]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002

[9]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[10]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

[18]

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

[19]

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

[20]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]