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.
