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Symmetry and non-existence of solutions to an integral system
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China |
$\left\{ \begin{matrix} {{u}_{i}}(x)=\int_{{{\mathbf{R}}^{n}}}{\frac{1}{|x-y{{|}^{n-\alpha }}}}{{f}_{i}}(u(y))dy,\ \ x\in {{\mathbf{R}}^{n}},\ \ i=1,\cdots ,m, \\ 0<\alpha <n,\text{and }\ u(x)=({{u}_{1}}(x),{{u}_{2}}(x),\cdots ,{{u}_{m}}(x)). \\\end{matrix} \right.\ \ \ \ \ \ \left( 1 \right)$ |
$ f_i(u)∈ C^1(\mathbf{R^m_+})\bigcap$ |
$ C^0(\mathbf{\overline{R^m_+}})(i = 1,2,···,m)$ |
$ β_{i}$ |
$ 0<β_{i} ≤q \frac{n+α}{n-α}$ |
$ u_1, u_2, ···, u_m$ |
$ u = (u_1,u_2,···,u_m)$ |
$ u$ |
$ f_i(u) = \sum_{r = 1}^{k}f_{ir}(u),$ |
$ f_{ir}(u)$ |
$ β_{ir}, r = 1,2,···,k$ |
$ 0 <β_{ir} ≤q \frac{n+α}{n-α}$ |
References:
[1] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive soltions of nonlinear elliptic equaions in $ {\mathbf{R}}^n$,
Mathematical Analysis And Applications, Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[2] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
|
[3] |
A. Chang and P. Yang,
On uniqueness of solutions of n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12.
|
[4] |
W. Chen and C. Li,
Classification of positive solutions for nonlinear differential and integral systems with critial exponents, Acta Math Sci., 29 (2009), 949-960.
|
[5] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
|
[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.
|
[7] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
|
[8] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
|
[9] |
W. Chen and C. Li,
A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
|
[10] |
Y. Fang and W. Chen,
A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
|
[11] |
C. Jin and C. Li,
Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
|
[12] |
J. Serrin,
A Symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
|
[13] |
C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
|
[14] |
C. Li and L. Ma,
Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2007), 1049-1057.
|
[15] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a nonlinear system involving the fractional laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
|
show all references
References:
[1] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive soltions of nonlinear elliptic equaions in $ {\mathbf{R}}^n$,
Mathematical Analysis And Applications, Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[2] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
|
[3] |
A. Chang and P. Yang,
On uniqueness of solutions of n-th order differential equation in conformal geometry, Math. Res. Letters, 4 (1997), 1-12.
|
[4] |
W. Chen and C. Li,
Classification of positive solutions for nonlinear differential and integral systems with critial exponents, Acta Math Sci., 29 (2009), 949-960.
|
[5] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
|
[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.
|
[7] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
|
[8] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
|
[9] |
W. Chen and C. Li,
A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.
|
[10] |
Y. Fang and W. Chen,
A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.
|
[11] |
C. Jin and C. Li,
Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.
|
[12] |
J. Serrin,
A Symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
|
[13] |
C. Li,
Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.
|
[14] |
C. Li and L. Ma,
Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Anal., 40 (2007), 1049-1057.
|
[15] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a nonlinear system involving the fractional laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
|
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