We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R+n,{u（x） =2/（nωn）∫?R+n（xnf（v（y）））/（|x- y|n）dy, x ∈R+n,v（y） =2/（nωn）∫R+n（xng（u（x）））/（...We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R+n,{u（x） =2/（nωn）∫?R+n（xnf（v（y）））/（|x- y|n）dy, x ∈R+n,v（y） =2/（nωn）∫R+n（xng（u（x）））/（|x- y|n）dx, y ∈?R+n,where n 3, ωn is the volume of the unit ball in Rn. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Hang et al.（2008）.With natural structure conditions on f and g, we classify the positive solutions of the above system based on the method of moving spheres in integral form and the inequality mentioned above.展开更多
We investigate the nonnegative solutions of the system involving the fractional Laplacian：{（-△）^αui（x）=fi（u）,x∈R^n,i=1,2,…,m, u（x）=（u1（x）,u2（x）,……,um（x））,where 0 〈 α 〈 1, n 〉 2, fi（u）, 1 4...We investigate the nonnegative solutions of the system involving the fractional Laplacian：{（-△）^αui（x）=fi（u）,x∈R^n,i=1,2,…,m, u（x）=（u1（x）,u2（x）,……,um（x））,where 0 〈 α 〈 1, n 〉 2, fi（u）, 1 4 ≤ 4 ≤m, are real-valued nonnegative functions of homogeneous degree Pi ≥0 and nondecreasing with respect to the independent variables ul, u2,..., urn. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x0 if Pi = （n ＋ 2α）/（n- 2α） for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1〈pi〈（n＋2α）/（n-2α） for all 1≤i≤m.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11571268)Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2014JM1021)
文摘We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R+n,{u（x） =2/（nωn）∫?R+n（xnf（v（y）））/（|x- y|n）dy, x ∈R+n,v（y） =2/（nωn）∫R+n（xng（u（x）））/（|x- y|n）dx, y ∈?R+n,where n 3, ωn is the volume of the unit ball in Rn. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Hang et al.（2008）.With natural structure conditions on f and g, we classify the positive solutions of the above system based on the method of moving spheres in integral form and the inequality mentioned above.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant No. 11171266）.
文摘We investigate the nonnegative solutions of the system involving the fractional Laplacian：{（-△）^αui（x）=fi（u）,x∈R^n,i=1,2,…,m, u（x）=（u1（x）,u2（x）,……,um（x））,where 0 〈 α 〈 1, n 〉 2, fi（u）, 1 4 ≤ 4 ≤m, are real-valued nonnegative functions of homogeneous degree Pi ≥0 and nondecreasing with respect to the independent variables ul, u2,..., urn. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x0 if Pi = （n ＋ 2α）/（n- 2α） for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1〈pi〈（n＋2α）/（n-2α） for all 1≤i≤m.
