Aleks Jevnikar, Andrea Malchiodi, Ruijun Wu
We are concerned with super-Liouville equations on the two sphere, which have variational structure with a strongly-indefinite functional. We prove the existence of non-trivial solutions by combining the use of Nehari manifolds, balancing conditions and bifurcation theory.
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The two-point boundary value problem for the one-dimensional Liouville type equation is considered. In this paper, a symmetry-breaking result is obtained by using the Morse index.
Morse index and symmetry-breaking bifurcation of positive solutions to the one-dimensional Liouville type equation with a step function weight
本文针对带阶跃函数权重的一维Liouville型方程，计算正偶对称解的Morse指标，并证明从对称破缺分歧点出发存在无界的正非偶对称解连通集。
We prove the existence of a family of non-trivial solutions of the Liouville equation in dimensions two and four with infinite volume. These solutions are perturbations of a finite-volume solution of the same equation in one dimension less. In particular, they are periodic in one variable and decay linearly to $-\infty$ in the other variables. In dimension two, we also prove that the periods are arbitrarily close to $πk, k \in \mathbb{N}$ (from the positive side). The main tool we employ is bifurcation theory in weighted Hölder spaces.
In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-Δ)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-Δ)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the functions $h$ and $f$ are nondecreasing. The main feature is that the function $h$ changes sign in $\mathbb{R}$, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
Bifurcation of Tetrahedral Non-Zonal Flows in the 2D Euler Equations on a Rotating Sphere
研究旋转球面上二维欧拉方程静止解引发有限振幅非区流的出现，通过限制拉普拉斯-贝尔特里特征空间到四面体对称群不变子空间，获得标量李雅普诺夫-施密特约简，分析四种非线性模型并证明分支拓扑由非线性奇点的奇偶性及质量守恒决定。
We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The t…
Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -Δu + u =μ_1 u^3 + βuv^2- κv, -Δv + v =μ_2 v^3 + βu^2v- κu, u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N), \end{array} \right. \end{equation*}where $μ_1, μ_2$ are positive and fixed, $κ$ and $β$ are linear and nonlinear coupling parameters respectively. We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system. Then using the positive and non-dege…
We study the triviality of the solutions of weighted superlinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a Liouville--type theorem for solutions bounded from below with nonnegative initial data, under an integral growth condition on the weight.
Super-Liouville equation with a spinorial Yamabe type term
本文研究带有旋量Yamabe型项的超Liouville方程，为爆破序列建立精细定性性质，证明旋量与函数部分的能量恒等式并计算局部质量，揭示了由非线性项导致的两种奇性新现象。
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blow up analysis.
We are concerned with a super-Liouville equation on compact surfaces with genus larger than one, obtaining the first non-trivial existence result for this class of problems via min-max methods. In particular we make use of a Nehari manifold and, after showing the validity of the Palais-Smale condition, we exhibit either a mountain pass or linking geometry.
Super-Liouville equation with a Dirichlet-chiral boundary
本文证明在Dirichlet-手性边界条件下，超Liouville方程不存在爆破，解决了其紧性问题。
We address a new numerical scheme based on a class of machine learning methods, the so-called Extreme Learning Machines with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous mixed boundary conditions, and, (b) the one and two-dimensional Liouville-Bratu-Gelfand PDEs with homogeneous…
Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance
本文针对无尺度不变性的超线性抛物问题，利用椭圆刘维尔定理、能量估计和分量比例化约化方法建立全空间与半空间中的刘维尔型定理，并将其应用于普适奇异性与衰减估计。
We consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (i) can inviscid instability occur? (ii) can there occur viscous instability not detected by inviscid theory? (iii) can there occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping instability"? and, perhaps most important from a practical point…
We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is $F(x,D^2u)\geq H_i(x,u,Du)$ in $\mathbb{R}^N$, where $H_i$ has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases $H_1(u,Du)=u^q+|Du|^γ$, $H_2(u,Du)=u^q|Du|^γ$ and $H_3(x,u,Du)=\pm u^q|Du|^γ-b(x)\cdot Du$, where $q\geq0$, $γ>1$ and $b$ is a suitable velocity field. Several counterexamples and open problems are tho…
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 use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.
In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability …
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation $$u_t-Δu=u^p\quad\hbox{in}\quad {R}^n\times{R}, \qquad p>1, $$ the nonexistence of positive classical solutions in the subcritical range $p(n-2)<n+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on $p$ or deal with a special cl…
We study solutions and supersolutions of homogeneous and nonhomogeneous $\mathcal{A}$-harmonic equations with nonstandard growth in $\mathbb{R}^n$. Various Liouville-type theorems and nonexistence results are proved. The discussion is illustrated by a number of examples.
We establish a Liouville comparison principle for entire sub- and super-solutions of the equation $(\ast)$ $w_t-Δ_p (w) = |w|^{q-1}w$ in the half-space ${\mathbb S}= {\mathbb R}^1_+\times {\mathbb R}^n$, where $n\geq 1$, $q>0$ and $ Δ_p (w):={div}_x(|\nabla_x w|^{p-2}\nabla_x w)$, $1<p\leq 2$. In our study we impose neither restrictions on the behaviour of entire sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on the behavior of them or any of their partial derivatives at infinity. We prove that if $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respectively, an entir…
Existence of Solutions to a super-Liouville equation with Boundary Conditions
本文研究带边界且欧拉示性数为负的紧致黎曼曲面上超刘维尔方程解的存在性，通过引入加权 Dirac 算子并构造 Nehari 流形，利用极小极大理论证明了非平凡解的存在。
We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.
We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation $(\ast)$ $w_t-Δ_p (w) = |w|^{q-1}w$ in the half-space ${\mathbb S}= {\mathbb R}^1_+\times {\mathbb R}^n$, where $n\geq 1$, $q>0$ and $ Δ_p (w):={div}_x(|\nabla_x w|^{p-2}\nabla_x w)$, $1<p\leq 2$. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respe…
In this paper we consider the problem $-Δu=|x|^α F(u)$ in $R^N$, with $α>0$ and $N\ge3$. Under some assumptions on $F$ we deduce the existence of nonradial solutions which bifurcate from the radial one when $α$ is an even integer.
We consider the elliptic quasilinear equation --$Δ$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a p…
In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlineari…
Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions
本文研究二维单位球中一般超临界非线性Gelfand问题，证明小参数下不稳定解不存在、径向奇异解存在及有限Morse指标解一致有界，从而得出分岔曲线具有无穷多个转折点，并阐明分岔结构仅由非线性超临界性决定。
A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $λ-$direction and bifurcates from infinity at $λ=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.
Weighted Liouville Theorems and A Priori Bounds for Indefinite Superlinear Equations with Pucci's Maximal Operator
本文针对带有Pucci极大算子的不定超线性方程，通过证明Liouville型定理，建立其正解集的一致有界性。
A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient
本文研究了梯度具有超二次增长的椭圆型方程$-Δu = u^q|∇u|^p$（$p≥2$），利用次调和函数球面平均的单调性与局部Bernstein论证，证明了正有界解必为常数的Liouville型定理，解决了相关文献遗留的$p≥2$情形问题，并推广至方程组。
We study radial solutions of the semilinear elliptic equation $Δu+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $Δu+u^p=0$ and $Δu+…
In \cite{LWZ}, we establish Liouville-type theorems and decay estimates for solutions of a class of high order elliptic equations and systems without the boundedness assumptions on the solutions. In this paper, we continue our work in \cite{LWZ} to investigate the role of boundedness assumption in proving Liouville-type theorems for fully nonlinear equations. We remove the boundedness assumption of solutions which was required in the proof of Liouville-type theorems for fully nonlinear elliptic equations or systems in half spaces. We also prove the Liouville-type theorems for supersolutions of…
Some new global results are given about solutions to the boundary value problem for the Euler-Lagrange equations for the Ginzburg-Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range there is a branch of asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also, simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These proofs require no detailed asymptotics for the solutions of the linear equations. Finally, an error in Odeh's work on this problem is discussed.
Compactness of Palais-Smale sequences with controlled Morse Index for a Liouville type functional
本文证明闭曲面上 Liouville 型泛函的有界 Morse 指数 Palais-Smale 序列具有预紧性，并据此给出超临界平均场方程解存在性的新证明及奇异情形推广。
Liouville theorems and universal estimates for superlinear elliptic problems without scale invariance
本文将刘维尔型定理应用于无尺度不变性的椭圆系统以获得普适奇性和衰减估计，提出了基于 Pohozaev 恒等式等方法证明新定理，并综述改进了现有证明方法。
In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for systems of equations involving potential terms and the fractional Laplacian in an exterior domain. Such problems arise in the analysis of a priori estimates of solutions. The class of problems we consider in this article is quite general compared to the literature. The main ingredient for our proofs is the hitting time estimates for the symmetric $α$-stable process and probabilistic representation of the super-solutions.
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