February  2016, 36(2): 1125-1141. doi: 10.3934/dcds.2016.36.1125

Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

1. 

Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033, United States

2. 

Department of Mathematics, Yeshiva University, New York, NY 10033

3. 

Department of Applied Mathematics, Northwestern Polytechnical University, Xian 710072, Shanxi, China

4. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, China

Received  July 2014 Revised  February 2015 Published  August 2015

In this paper, we consider the following system of pseudo-differential nonlinear equations in $R^n$ \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u_i (x)= f_i( u_1(x), \cdots u_m(x)), & i=1, \cdots, m, \\ u_i \geq 0 , & i=1, \cdots, m,             (1) \end{array} \right. \label{b1} \end{equation} where $\alpha$ is any real number between $0$ and $2$.
    We obtain radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.
    To this end, we first establish the equivalence between (1) and the corresponding integral system $$ \left\{\begin{array}{ll} u_i(x) = \int_{R^n} \frac{c_n}{|x-y|^{n-\alpha}} f_i( u_1(y), \cdots, u_m(y)), & i=1, \cdots, m, \\ u_i(x) \geq 0, & i=1, \cdots, m. \end{array} \right. $$ A new idea is introduced in the proof, which may hopefully be applied to many other problems. Combining this equivalence with the existing results on the integral system, we obtained much more general results on the qualitative properties of the solutions for (1).
Citation: Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
References:
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D. Applebaum, Lévy Processes and Stochastic Calculus,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781. Google Scholar

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C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39. doi: 10.1017/S0308210511000175. Google Scholar

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

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K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes,, Illinois J. Math., 46 (2002), 541. Google Scholar

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J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications,, Physics reports, 195 (1990), 127. doi: 10.1016/0370-1573(90)90099-N. Google Scholar

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L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. Math., 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. Google Scholar

[7]

L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems,, Disc. Cont. Dyna. Sys., 33 (2013), 3937. doi: 10.3934/dcds.2013.33.3937. Google Scholar

[8]

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

[9]

W. Chen and C. Li, Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar

[10]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff.Equa.Dyn.Sys., (2010). Google Scholar

[11]

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

[12]

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

[13]

P. Constantin, Euler equations, navier-stokes equations and turbulence,, in Mathematical Foundation of Turbulent Viscous Flows, 1871 (2006), 1. doi: 10.1007/11545989_1. Google Scholar

[14]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[15]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Advances in Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[16]

I. Capuzzo-Dolcetta and A. Cutri, On the Liouville property for sub-Laplacians,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 239. Google Scholar

[17]

L. Dupaigne and Y. Sire, A Liouville theorem for nonlocal elliptic equations,, Symmetry for elliptic PDEs Contemp. Math., 528 (2010), 105. doi: 10.1090/conm/528/10417. Google Scholar

[18]

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

[19]

Y. Fang and J. Zhang, Nonexistence of positive solution for an integral equation on a half-space $R^n_+$,, Comm. Pure and Applied Analysis, 12 (2013), 663. doi: 10.3934/cpaa.2013.12.663. Google Scholar

[20]

T. Kulczycki, Properties of Green function of symmetric stable processes,, Probability and Mathematical Statistics, 17 (1997), 339. Google Scholar

[21]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian,, Nonlinear Analysis, 75 (2012), 3036. doi: 10.1016/j.na.2011.11.036. Google Scholar

[22]

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

[23]

E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$,, Differential & Integral Equations, 9 (1996), 465. Google Scholar

[24]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar

[25]

E. M. Stein, Singular Integrals and Differentiablity Properties of Funciotns,, Princeton University Press, (1970). Google Scholar

[26]

P. Stinga and C. Zhang, Harnack's inequality for fractional nonlocal equations,, Disc. Cont. Dyn. Sys., 33 (2013), 3153. doi: 10.3934/dcds.2013.33.3153. Google Scholar

[27]

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

[28]

M. Zhu, Liouville theorems on some indefinite equations,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 649. doi: 10.1017/S0308210500021569. Google Scholar

[29]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian,, , (). Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781. Google Scholar

[2]

C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39. doi: 10.1017/S0308210511000175. Google Scholar

[3]

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

[4]

K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes,, Illinois J. Math., 46 (2002), 541. Google Scholar

[5]

J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications,, Physics reports, 195 (1990), 127. doi: 10.1016/0370-1573(90)90099-N. Google Scholar

[6]

L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. Math., 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. Google Scholar

[7]

L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems,, Disc. Cont. Dyna. Sys., 33 (2013), 3937. doi: 10.3934/dcds.2013.33.3937. Google Scholar

[8]

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

[9]

W. Chen and C. Li, Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar

[10]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff.Equa.Dyn.Sys., (2010). Google Scholar

[11]

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

[12]

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

[13]

P. Constantin, Euler equations, navier-stokes equations and turbulence,, in Mathematical Foundation of Turbulent Viscous Flows, 1871 (2006), 1. doi: 10.1007/11545989_1. Google Scholar

[14]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[15]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Advances in Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[16]

I. Capuzzo-Dolcetta and A. Cutri, On the Liouville property for sub-Laplacians,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 239. Google Scholar

[17]

L. Dupaigne and Y. Sire, A Liouville theorem for nonlocal elliptic equations,, Symmetry for elliptic PDEs Contemp. Math., 528 (2010), 105. doi: 10.1090/conm/528/10417. Google Scholar

[18]

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

[19]

Y. Fang and J. Zhang, Nonexistence of positive solution for an integral equation on a half-space $R^n_+$,, Comm. Pure and Applied Analysis, 12 (2013), 663. doi: 10.3934/cpaa.2013.12.663. Google Scholar

[20]

T. Kulczycki, Properties of Green function of symmetric stable processes,, Probability and Mathematical Statistics, 17 (1997), 339. Google Scholar

[21]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian,, Nonlinear Analysis, 75 (2012), 3036. doi: 10.1016/j.na.2011.11.036. Google Scholar

[22]

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

[23]

E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$,, Differential & Integral Equations, 9 (1996), 465. Google Scholar

[24]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar

[25]

E. M. Stein, Singular Integrals and Differentiablity Properties of Funciotns,, Princeton University Press, (1970). Google Scholar

[26]

P. Stinga and C. Zhang, Harnack's inequality for fractional nonlocal equations,, Disc. Cont. Dyn. Sys., 33 (2013), 3153. doi: 10.3934/dcds.2013.33.3153. Google Scholar

[27]

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

[28]

M. Zhu, Liouville theorems on some indefinite equations,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 649. doi: 10.1017/S0308210500021569. Google Scholar

[29]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian,, , (). Google Scholar

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