In this paper, we consider systems of fractional Laplacian equations in ]I^n with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We ...In this paper, we consider systems of fractional Laplacian equations in ]I^n with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].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Partially supported by NSFC(11571233)NSF DMS-1405175+1 种基金NSF of Shanghai16ZR1402100China Scholarship Council
文摘In this paper, we consider systems of fractional Laplacian equations in ]I^n with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].
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.
