We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi （i = 1, 2） satisfy some suitable assumptions. Due to equivalence between the ...We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi （i = 1, 2） satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1 ＋ p2= q1 ＋ q2 = n＋α/n-α, we classify the solutions of the PDEs system.展开更多
In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 <...In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).展开更多
The aim of the paper is to study the properties of positive classical solutions to the fractional Laplace equation with the singular term.Using the extension method,we prove the nonexistence and symmetric of solutions...The aim of the paper is to study the properties of positive classical solutions to the fractional Laplace equation with the singular term.Using the extension method,we prove the nonexistence and symmetric of solutions to the singular fractional equation.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11571268)the foundation of Xi’an University of Finance and Economics(Grant No.12XCK07)
文摘We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,where 0 〈α〈 n, pi and qi （i = 1, 2） satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1 ＋ p2= q1 ＋ q2 = n＋α/n-α, we classify the solutions of the PDEs system.
基金supported by NSFC(11761082)Yunnan Province,Young Academic and Technical Leaders Program(2015HB028)
文摘In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
基金supported by the NNSF of China(11371056)partly supported by the NNSF of China(11501021)+1 种基金the China Postdoctoral Science Foundation(2013M540057)partly supported by Scientific Research Fund of Jiangxi Provincial Education Department(GJJ160797)
文摘In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).
基金supported by the Key Scientific Research Project of the Colleges and Universities in Henan Province(No.22A110013)the Key Specialized Research and Development Breakthrough Program in Henan Province(No.222102310265)+1 种基金the Natural Science Foundation of Henan Province of China(No.222300420499)the Cultivation Foundation of National Natural Science Foundation of Huanghuai University(No.XKPY-202008).
文摘The aim of the paper is to study the properties of positive classical solutions to the fractional Laplace equation with the singular term.Using the extension method,we prove the nonexistence and symmetric of solutions to the singular fractional equation.
基金Supported by Chinese National Science Fund for Distinguished Young Scholars(No.11101319,No.11201081,No.11202035)the Foundation of Shaanxi Statistical Research Center(No.13JD04)the Foundation of Shaanxi Province Education Department(No.14JK1276)
