In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R^+_n:{u(x)=c_n,a∫_?R_+~n(x_n^(1-a_v)(y)/|x-y|^(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n^...In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R^+_n:{u(x)=c_n,a∫_?R_+~n(x_n^(1-a_v)(y)/|x-y|^(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n^(1-a_uθ)(x)/|x-y|^(n-a))dx,y∈ ?R_+~n,where n 2, 2-n < a < 1, κ, θ > 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.展开更多
We study positive solutions to the following higher order SchrSdinger system with Dirichlet boundary conditions on a half space： where a is any even number between 0 and n. This PDE system is closely related to the i...We study positive solutions to the following higher order SchrSdinger system with Dirichlet boundary conditions on a half space： where a is any even number between 0 and n. This PDE system is closely related to the integral system where G is the corresponding Green＇s function on the half space. More precisely, we show that every solution to （0.2） satisfies （0.1）, and we believe that the converse is also true. We establish a Liouville type theorem the non-existence of positive solutions to （0.2） under a very weak condition that u and v are only locally integrable. Some new ideas are involved in the proof, which can be applied to a system of more equations.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. 2017JQ1022)the Fundamental Research Funds for the Central Universities (Grant No. GK201802015)
文摘In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R^+_n:{u(x)=c_n,a∫_?R_+~n(x_n^(1-a_v)(y)/|x-y|^(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n^(1-a_uθ)(x)/|x-y|^(n-a))dx,y∈ ?R_+~n,where n 2, 2-n < a < 1, κ, θ > 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.
基金supported by China Scholarship Council(Grant No.201206060010)
文摘We study positive solutions to the following higher order SchrSdinger system with Dirichlet boundary conditions on a half space： where a is any even number between 0 and n. This PDE system is closely related to the integral system where G is the corresponding Green＇s function on the half space. More precisely, we show that every solution to （0.2） satisfies （0.1）, and we believe that the converse is also true. We establish a Liouville type theorem the non-existence of positive solutions to （0.2） under a very weak condition that u and v are only locally integrable. Some new ideas are involved in the proof, which can be applied to a system of more equations.
