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On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations
Radial symmetry of nonnegative solutions for nonlinear integral systems
School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China |
$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0<\alpha<n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $ |
$ 0<a_i/2<\alpha $ |
$ f_i(u) $ |
$ 1\leq i\leq m $ |
$ u_1 $ |
$ u_2 $ |
$ \cdots $ |
$ u_m $ |
$ u = (u_1,u_2,\cdots,u_m) $ |
$ f_i $ |
References:
[1] |
J. Busca and R. Manásevich,
A Liouville-type theorem for Lane-Emden systems, Indiana. Univ. Math. J., 51 (2002), 37-51.
|
[2] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure. Appl. Math., 42 (1989), 271-297.
doi: 10.1002/3160420304. |
[3] |
W. Chen and C. Li,
Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta. Math. Scie., 29B (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[4] |
W. Chen and C Li,
Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure. Appl. Anna., 12 (2013), 2497-2514.
doi: 10.3934/2013.12.2497. |
[5] |
D. G. de Figueiredo and P. L. Felmer,
A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.
|
[6] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbb{R}^n$, collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981., |
[7] |
D. Li, P. Niu and R. Zhuo,
Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.
doi: 10.1016/2014.11.029. |
[8] |
Y. Lv and C. Zhou,
Symmetry for an integral system with general nonlinearity, Disc. Cont. Dyna. Syst., 39 (2019), 1533-1543.
doi: 10.3934/dcds.2018121. |
[9] |
E. Mitidieri,
Non-existence of positive solutions of semilinear systems in $\mathbb{R}^n$, Differ. Int. Equ., 9 (1996), 465-479.
|
[10] |
J. Serrin and H. Zou,
Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1996), 369-380.
|
[11] |
J. Serrin and H. Zou,
The existence of positive solutions of elliptic Hamiltonian system, Commun. Partial Differ. Equ., 23 (1998), 577-599.
doi: 10.1080/03605309808821356. |
[12] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
show all references
References:
[1] |
J. Busca and R. Manásevich,
A Liouville-type theorem for Lane-Emden systems, Indiana. Univ. Math. J., 51 (2002), 37-51.
|
[2] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure. Appl. Math., 42 (1989), 271-297.
doi: 10.1002/3160420304. |
[3] |
W. Chen and C. Li,
Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta. Math. Scie., 29B (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[4] |
W. Chen and C Li,
Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure. Appl. Anna., 12 (2013), 2497-2514.
doi: 10.3934/2013.12.2497. |
[5] |
D. G. de Figueiredo and P. L. Felmer,
A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.
|
[6] |
B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbb{R}^n$, collected in the book Mathematical Analysis and Applications, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 1981., |
[7] |
D. Li, P. Niu and R. Zhuo,
Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.
doi: 10.1016/2014.11.029. |
[8] |
Y. Lv and C. Zhou,
Symmetry for an integral system with general nonlinearity, Disc. Cont. Dyna. Syst., 39 (2019), 1533-1543.
doi: 10.3934/dcds.2018121. |
[9] |
E. Mitidieri,
Non-existence of positive solutions of semilinear systems in $\mathbb{R}^n$, Differ. Int. Equ., 9 (1996), 465-479.
|
[10] |
J. Serrin and H. Zou,
Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1996), 369-380.
|
[11] |
J. Serrin and H. Zou,
The existence of positive solutions of elliptic Hamiltonian system, Commun. Partial Differ. Equ., 23 (1998), 577-599.
doi: 10.1080/03605309808821356. |
[12] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
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