In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of St...In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.展开更多
Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider...Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..展开更多
In this paper, we establish a Liouville-type theorem for a system of higher-order parabolic inequalities by using the method of test functions and an integral estimate. As an application, we observe the Fujita blow-up...In this paper, we establish a Liouville-type theorem for a system of higher-order parabolic inequalities by using the method of test functions and an integral estimate. As an application, we observe the Fujita blow-up phenomena for the corresponding parabolic system, which in particular fills up the gap in the recent result of Pang et. al. （Existence and nonexistence of global solutions for a higher-order semilinear parabolic system, Indiana Univ. Math. J., 55（2006）, 1113-1134）. Moreover, the importance of this observation is that we do not impose any regularity assumption on the initial data.展开更多
We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically zero.This is an extension of previous results, all of ...We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically zero.This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ▽u, ▽H∈L^2(R^3), i.e., finite Dirichlet integral.展开更多
In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space(X, dX)with curvature bounded above by a constant κ(κ 0) in the sense of Alexandrov. As a direct application,it gives some Lio...In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space(X, dX)with curvature bounded above by a constant κ(κ 0) in the sense of Alexandrov. As a direct application,it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng(1980) and Choi(1982) to harmonic maps into singular spaces.展开更多
基金National Natural Science Foundation of China(Nos.11731001,11971400,12101619)Guangdong Basic and Applied Basic Research Foundation(No.2020A1515011019)Science and Technology Projects of Guangzhou(No.202102020743)。
文摘In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.
基金Supported by the National Natural Science Foundation of China (11171254, 11271209)
文摘Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..
基金Supported by the Scientific Research Fund of Sichuan Provincial Education Department (Grant No.09ZB081)the Key Scientific Research Foundation of Xihua University (Grant No.Z0912611)+1 种基金Sichuan Youth Science & Technology Foundation (Grant No.2011JQ0003)the Fundamental Research Funds for the Central Universities
文摘In this paper, we establish a Liouville-type theorem for a system of higher-order parabolic inequalities by using the method of test functions and an integral estimate. As an application, we observe the Fujita blow-up phenomena for the corresponding parabolic system, which in particular fills up the gap in the recent result of Pang et. al. （Existence and nonexistence of global solutions for a higher-order semilinear parabolic system, Indiana Univ. Math. J., 55（2006）, 1113-1134）. Moreover, the importance of this observation is that we do not impose any regularity assumption on the initial data.
基金supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]
文摘We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically zero.This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ▽u, ▽H∈L^2(R^3), i.e., finite Dirichlet integral.
基金supported by National Natural Science Foundation of China (Grant No. 11521101)supported by National Natural Science Foundation of China (Grant No. 11571374)+1 种基金National Program for Support of Top-Notch Young Professionalssupported by the Academy of Finland
文摘In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space(X, dX)with curvature bounded above by a constant κ(κ 0) in the sense of Alexandrov. As a direct application,it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng(1980) and Choi(1982) to harmonic maps into singular spaces.
