含Grushin算子的加权椭圆系统稳定解的Liouville定理

考虑含Grushin算子的加权椭圆系统正稳定解的Liouville定理.首先建立这类椭圆系统稳定解的均值形式的判定准则,再建立新的比较原则,联合积分估计和Bootstrap迭代方法建立了权数指数不相等和指数为超临界时椭圆系统正稳定解的Liouville定理.     

加权的退化椭圆系统稳定解的Liouville定理

该文研究了加权的退化椭圆系统■其中Δ_Gu=Δ_xu+(a+1)²|x|^2aΔ_yu是Grushin算子,α,β≥0,q>1,ω(x)=(1+‖x‖⁽2(α+1)))⁽β/2(α+1)).超临界指数正稳定解的Liouville定理被建立.     

含多调和延拓算子的积分方程组的Liouville型定理

本文研究上半空间一类含有多调和延拓算子的积分方程组正解的分类问题.在某些自然结构假设下,利用积分形式的移动球面法和上半空间的积分不等式,获得了该积分方程组正解的Liouville型定理,推广了已有的结果.     

一类含多奇性项的Grushin型算子方程解的渐近性质

本文研究了一类含多个奇性项的Grushin型算子方程非平凡解的渐近性质问题.当方程的非线性项满足临界指数增长条件时,利用Moser迭代方法和分析技巧,获得了方程的非平凡解在奇点处的渐近性质,推广了Laplace算子的相关结果.     

三维稳态Navier-Stokes-Poission方程的Liouville型定理

本文研究了三维稳态Navier-Stokes-Poission方程的Liouville型定理.利用能量方法,证明了如果光滑解(ρ, u,Φ)满足一些合适的条件,则u=0.本文的结果推广了Chae的结果(Nonlinearity, 2012, 25(5):1345–1349)到Lorentz空间.     

一类Grushin亚椭圆算子的特征集

研究了与亚椭圆算子2x x 2β2y相对应的G rush in平面上余维数为1的子流形S的特征集C(S)在对应的G rush in度量下的H ausdorff维数和H ausdorff测度.     

带自由边界调和映射的Liouville型定理

本文给出一类带由边界的调和映射的Liouville型定理,这种类型的定理在微分几何的一些问题中有十分重要的应用.我们通过对调和映射的能量选取特殊的变分族,得到任意从半空间的简单流形到一黎曼流形的带自由边界的调和映射在如果满足适当的条件(见定理)必为常值映射的结果.     

非线性Hénon方程解的Liouville定理

该文研究了二阶和四阶非线性Henon-Lane-Emden方程有限Morse指标解的Liouville定理.利用一种新方法,即使用单调公式、Pohozaev恒等式和doubling引理等相结合证明了其结果.     

H型群上的非线性Liouville定理

对H型群上的次线性方程ΔGu up=0进行了研究,其中ΔG是对应的sublaplace算子．通过将移动平面方法推广到H型群上,证明了在一定条件下的p和函数u,次线性方程ΔGu up=0的唯一解是u≡0．     

H型群上的非线性Liouville定理

对H型群上的次线性方程△Gu+up=0进行了研究,其中△G是对应的sublaplace算子.通过将移动平面方法推广到H型群上,证明了在一定条件下的p和函数u,次线性方程△Gu+up=0的唯一解是u≡0.     

Liouville-type theorem for nonlinear elliptic equations involving p-Laplace–type Grushin operators

In this article, we prove the Liouville-type theorem for stable solutions of weighted p-Laplace–type Grushin equations (1) and (2) where p ≥ 2, q>0 and are nonnegative functions satisfying and as ‖z‖_G ≥ R₀ with p−N_γ<b<θ+p, R₀,C_i(i=1,2) are some positive constants. ∇_G=(∇_x,(1+γ)|x|^γ∇_y),γ ≥ 0, and The results hold true for N_γ<μ₀(p,b,θ) in 1 and q>q_c(p,N_γ,b,θ) in 2 . Here, μ₀ and q_c are new exponents, which are always larger than the classical critical ones and depend on the parameters p,b and θ. N_γ=N₁+(1+γ)N₂ is the homogeneous dimension of      

Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

Liouville theorems for the weighted Lane–Emden equation with finite Morse indices

In this paper, we study the nonexistence result for the weighted Lane–Emden equation: (0.1) and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: (0.2) where f(|x|) and g(|x|) are the radial and continuously differential functions, is an upper half space in , and . Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems 0.1 and 0.2 under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.     

On Liouville type theorems for fully nonlinear elliptic equations with gradient term

In this paper, we prove a Hadamard property and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with a gradient term, both in the whole space and in an exterior domain.     

Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

In this work, we utilize the existing Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, together with the telescoping series method, to establish an observability inequality from measurable subsets in time‐space variable for the parabolic equation with Grushin operator in some multidimension domains. We can apply this observability inequality to show the bang–bang property for both time optimal and norm optimal control problems for this kind of singular parabolic equation. Copyright © 2016 John Wiley & Sons, Ltd.