Xi-nan Ma, Qianzhong Ou
We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral estimate, which deduced from a generalized formula of that found by Jerison and Lee.
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We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
A New proof of Liouville type theorems for a class of semilinear elliptic equations
本文研究欧氏空间或具非负Ricci曲率的闭流形上的典型半线性椭圆方程，利用不变张量法构造的关键积分恒等式结合积分估计，重新证明了若干经典Liouville定理。
We prove a Liouville type result for bounded, entire solutions to a class of variational semilinear elliptic systems, based on the growth of their potential energy over balls with growing radius. Important special cases to which our result applies are the Ginzburg-Landau system and systems that arise in the study of multi-phase transitions.
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ Δu+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.
Monotonicity and Liouville-type theorems for semilinear elliptic problems in the half space
该研究探讨半空间中带齐次Dirichlet边界条件的半线性椭圆方程正解的性质，证明有限条带方向有界的正解沿边界法向严格单调递增，并推导出Lane-Emden方程的新刘维尔型定理。
The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ Δw-\frac{1}{2} \langle x,\nabla w\rangle-\fracλ{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical or critical and $λ\leq 1$, then all bounded entire solutions are constant. Moreover, in the critical case, the same conclusion holds in the subclass of radial solutions provided that $n\geq 4$ and $λ\in \left[\frac{3 n}{2(n-1)},2\right]$.
We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are define…
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
Li-Yau Inequality and Liouville Property to a Semilinear Heat Equation on Riemannian Manifolds
本文利用非线性方程的 Li–Yau 型估计，研究紧与非紧 Riemann 流形上半线性热方程整解的 Liouville 性质，并简化证明 Yau 与 Gidas–Spruck 定理。
In this paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. Surprisingly, the critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions.
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liou…
In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions …
Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
本文通过引入线性算子构造微分不等式，研究半线性椭圆方程正解的梯度估计与Liouville型定理，并将参数条件推广至更广范围，获得局部解的普适估计。
We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$ in the half-space ${\mathbb S} = {\mathbb R}^1_+ \times \mathbb R^n$. Here $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i} [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are functions defined, measurable and locally bounded in $\mathbb S$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ …
A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.
Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term
针对带半线性项的各向异性 $p$-Laplace 方程，结合 Serrin 技巧与不变张量方法，建立正解、超/子解及稳定变号解的 Liouville 不存在性与分类结果。
A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source
本文研究含非线性梯度源的半线性热方程，通过逐点梯度估计、Gidas-Spruck 积分法和 doubling lemma 证明 Liouville 型定理并导出普适先验估计。
We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ in the whole space ${\mathbb E} = {\mathbb R} \times \mathbb R^n$. Here, $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}\right],$$ where $a_{ij}(t,x)$, $i,j=1,\dots,n$, are functions that are defined, measurable and locally bounded in $\mathbb E$, and such that $a_{ij}(t,x)=a_…
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furtherm…
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
Liouville type results for semilinear elliptic equation and inequality on Sasakian manifolds
本文在伪 Hermitian 流形上，利用广义 Jerison-Lee 公式和体积估计，分别证明了半线性椭圆方程 $\Delta_b u+F(u)=0$ 的 Liouville 定理及不等式 $\Delta_b u+F(u)\leq 0$ 正解的不存在性。
Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source
本文通过构造线性算子和辅助函数，利用最大值原理，对所有参数(p,q)建立了含非线性梯度源的半线性椭圆方程的局部梯度估计与Liouville型定理，并推广了已有结果。
Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical points. Using the result and a quite standard variational method, it is found that the equation $$ -Δ_{H^n} u=|u|^{p-1} u ~ x\in H^n $$ has infinite positive solutions. Our approach can also be applied to study more general nonlinear problems.
Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group
该研究针对Heisenberg群上由Heisenberg向量场构造的完全非线性方程，通过退化Hadamard三球面定理证明新的尖锐Liouville型性质，并将欧几里得框架下Pucci极值算子受扰情况的结果推广到Heisenberg群。
Liouville results for semilinear integral equations with conical diffusion
本文研究锥形区域上半线性非局部方程正上解的不存在性，证明在次临界指数范围内零解是唯一非负上解，并给出临界指数分界及全空间推广。
Semilinear damped wave equations with data from Sobolev spaces of negative order: the critical case in Euclidean setting and in the Heisenberg space
本文证明了在欧几里得空间和海森堡群中，具有临界非线性的半线性阻尼波动方程在初始数据满足小能量和负阶 Sobolev 消失条件时的全局存在性。
A Liouville theorem for the $2$-Hessian equation on the Heisenberg group
本文通过选取检验函数和分部积分，证明了海森堡群上$2$-Hessian方程的Liouville定理。
In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group $\mathbb{H}^{n}$ involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As a special case, we obtain the Liouville theorem for the Lane-Emden system on the Heisenberg group $\mathbb{H}^{n}$, which also appears to be a new result in the literature.
We are concerned with solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we show that $u$ is independent of time. Our method of proof uses Serrin's sweeping principle, based on the strong maximum principle, applied to the linearized equation for $u_t$. Our result covers that of Poláčik and Yanagida […
In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-Δ_H$, where $Δ_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical…
In this paper, we establish Liouville theorems for the following system of elliptic differential inequalities $$ Δ_{\mathbb H}u^{m_1}+|η|_{\mathbb H}^{γ_1}|v|^p\leq0,$$ $$ Δ_{\mathbb H}v^{m_2}+|η|_{\mathbb H}^{γ_2}|u|^q\leq0,$$ on different unbounded open domains of Heisenberg group $\mathbb H$, including the whole space, and half space of $\mathbb H$. Here $p>m_2>0$, $q>m_1>0$.
Gradient estimate and Liouville theorem for a semilinear parabolic equation with variable coefficient
本文针对完备黎曼流形上带变系数的半线性抛物方程，通过Nash-Moser迭代建立正解的Li-Yau型梯度估计，进而推导古老解与永恒解的刘维尔定理，推广了相关经典结果。
Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group
本文研究了 Heisenberg 群上分数阶半线性热方程的可解性，给出了初值的充要条件，确定了 Fujita 指数并估计了解的寿命。
Under appropriate assumptions, we show that all bounded entire solutions to a class of semilinear elliptic systems are confined in a convex domain. Moreover, we prove a Liouville type theorem in the case where the domain is strictly convex. Our result represents an extension, under less regularity assumptions, of a recent result of Smyrnelis. We also provide several applications.
We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup + \inf$ type inequality and the quantized blowup mechanism.
We study the Lie point symmetries of semilinear Kohn-Laplace equations on the Heisenberg group H^1 and obtain a complete group classification of these equations.
We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre, as established in \cite{FGMT}. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space $\mathbb H^n\times \mathbbR^+$.
Liouville Theorem for Lane-Emden Equation on the Heisenberg Group
该研究针对Heisenberg群上的Lane-Emden方程，建立了稳定及紧集外稳定情形下的Liouville型定理，证明当p小于Joseph-Lundgren指数且不等于Sobolev指数时，0是唯一的紧集外稳定解。
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
Decay of mass for a semilinear heat equation on Heisenberg group
本文研究海森堡群上带时间依赖吸收的半线性热方程，证明非线性项与扩散效应在不同临界指数下决定大时间渐近行为。
Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group
本文在海森堡群上研究带强迫项的分数次亚拉普拉斯半线性热方程，确定其 Fujita 指数为 Q/(Q-2s)，并证明超临界情形全局解存在、次临界情形全局解不存在、临界情形存在有限时间爆破解，从而将经典 Fujita 现象推广到次黎曼非局部框架。
Appearance of Strauss-type exponent in semilinear wave equations with time-dependent speed of propagation
本文研究带时变传播速度与散射阻尼的半线性波动方程的小初值爆破现象，采用带Liouville–Green近似特殊解守恒量的检验函数法，得到次Strauss指数下的爆破结果，覆盖半线性广义Tricomi方程。
This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that …
Liouville theorem for subcritical nonlinear heat equation
研究亚临界非线性热方程的非负古老解，利用Li-Yau型估计和Melre-Zaag结果证明了该方程的利普希茨定理。
Optimal regularity & Liouville property for stable solutions to semilinear elliptic equations in $\mathbb R^n$ with $n\ge10$
本文证明了半线性椭圆方程稳定解在 $n\ge10$ 时的最优内部正则性，并由此推出尖锐的Liouville性质，回答了Cabré等人的开放问题。
In this paper, we prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R N , as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. We also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.
We consider entire solutions to $L u= f(u)$ in $\RR^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets.
We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $Δu=\nabla W(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, with $W:\mathbb{R}^m\to \mathbb{R}$, $m\geq 1$, nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution $u$). As an application, we can prove a Liouville type theorem under various assumptions.
We show that any Lie point symmetry of semilinear Kohn-Laplace equations on the Heisenberg group H^1 with power nonlinearity is a divergence symmetry if and only if the corresponding exponent assumes critical value.
Fractional semilinear damped wave equation on the Heisenberg group
本文研究海森堡群上分数阶半线性阻尼波动方程的柯西问题，推导了线性方程解及其导数的衰减估计，证明了不同条件下的全局适定性，并将结果应用于弱耦合系统的适定性分析。
In this paper, we study the asymptotic behavior as $x_1\to+\infty$ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at $x_1=0$ is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space~$\mathbb{R}^N$. The second one is based on the theory of dynamical …
Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order
研究Heisenberg群上半线性阻尼波动方程在负阶Sobolev初始数据下的Cauchy问题，确定临界指数，并证明超临界全局存在性、次临界有限时间爆破及寿命上下界。
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.
We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.
Let $0<α,β<2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \begin{equation*} \left\{\begin{array}{lll} (-\lap)^{α/2} u(x)=f(v(x)), & (-\lap)^{β/2} v(x)=g(u(x)), & \qquad x\in\mathbb{R}^n_+, u,v\geq0, & \qquad x\in\mathbb{R}^n\setminus\mathbb{R}^n_+. \end{array}\right. \end{equation*} Applying a direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove Liouville theorems of nonnegative solutions under some natural conditions on $f$ and $g$.
The Liouville equation with non-constant magnetic field is obtained as a limit in the Planck constant \hbar of the Heisenberg equation with the same magnetic field. The convergence is with respect to an appropriate semi-classical pseudo distance, and consequently with respect to the Monge-Kantorovich distance. Uniform estimates both in εand \hbar are proved for the specific 2D case of a magnetic vector potential of the form \frac {1} εx^{\bot}. As an application, an observation inequality for the Heisenberg equation with a magnetic vector potential is obtained. These results are a magnetic var…
This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solu…
We show which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries and we establish their respectives conservations laws.
Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds
本文在具有非负Ricci曲率的Riemannian流形上，通过严格证明亚临界Lane-Emden-Fowler方程正解的不存在性，扩展了Gidas-Spruck和Serrin-Zou的工作。
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