Symmetry and non-existence of solutions to an integral system

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May 2018, 17(3): 807-821. doi: 10.3934/cpaa.2018041

Symmetry and non-existence of solutions to an integral system

Yingshu Lü ^,

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

* Corresponding author

Received June 2017 Revised June 2017 Published January 2018

In this paper, we consider the nonnegative solutions of the following system of integral form:

$\left\{ \begin{matrix} {{u}_{i}}(x)=\int_{{{\mathbf{R}}^{n}}}{\frac{1}{|x-y{{|}^{n-\alpha }}}}{{f}_{i}}(u(y))dy,\ \ x\in {{\mathbf{R}}^{n}},\ \ i=1,\cdots ,m, \\ 0<\alpha <n,\text{and }\ u(x)=({{u}_{1}}(x),{{u}_{2}}(x),\cdots ,{{u}_{m}}(x)). \\\end{matrix} \right.\ \ \ \ \ \ \left( 1 \right)$

Here

$ f_i(u)∈ C^1(\mathbf{R^m_+})\bigcap$

$ C^0(\mathbf{\overline{R^m_+}})(i = 1,2,···,m)$

are real-valued functions, nonnegative, homogeneous of degree

$ β_{i}$

, where

$ 0<β_{i} ≤q \frac{n+α}{n-α}$

, and monotone nondecreasing with respect to the variables

$ u_1, u_2, ···, u_m$

. We show that the nonnegative solution

$ u = (u_1,u_2,···,u_m)$

is radially symmetric in the critical and subcritical case by method of moving planes in an integral form and

$ u$

must be zero in the subcritical case.

Futhermore, we consider the form of

$ f_i(u) = \sum_{r = 1}^{k}f_{ir}(u),$

where

$ f_{ir}(u)$

are real-valued homogeneous functions of various degrees

$ β_{ir}, r = 1,2,···,k$

and

$ 0 <β_{ir} ≤q \frac{n+α}{n-α}$

. We also show that the radial symmetry property of the nonnegative solution. Due to the homogeneous of degree can be different, the more intricate method is needed to deal with this difficulty.

Keywords: Integral system, nonnegative solutions, method of moving planes in an integral form, radial symmetry.

Mathematics Subject Classification: Primary: 45G05, 45G15, 45M99.

Citation: Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. Google Scholar
[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. Google Scholar
[7]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. Google Scholar
[8]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. Google Scholar
[9]	W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. Google Scholar
[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. Google Scholar
[11]	C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. Google Scholar
[12]	J. Serrin, A Symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. Google Scholar
[13]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. Google Scholar
[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. Google Scholar
[7]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. Google Scholar
[8]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. Google Scholar
[9]	W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564. Google Scholar
[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. Google Scholar
[11]	C. Jin and C. Li, Symmetry of solutions to some integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. Google Scholar
[12]	J. Serrin, A Symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. Google Scholar
[13]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. Google Scholar
[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. Google Scholar
[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. Google Scholar

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