Ali Hyder, Wen Yang
We study finite Morse index solutions to the non-local Gelfand-Liouville problem $$ (-Δ)^su=e^u\quad\mathrm{in}\quad \mathbb{R}^n,$$ for every $s\in(0,1)$ and $n>2s$. Precisely, we prove non-existence of finite Morse index solutions whenever the singular solution $$u_{n,s}(x)=-2s\log|x|+\log \left(2^{2s}\frac{Γ(\frac{n}{2})Γ(1+s)}{Γ(\frac{n-2s}{2})}\right)$$ is unstable.
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We give a quantization analysis to an elliptic system (Gelfand-Liouville type system) with Dirichlet condition. An application, we have a com-pactness result for an elliptic system with Lipschitz condition.
In this paper we provide a formal matched asymptotic analysis for large solutions to the Gelfand-Liouville problem in planar, doubly connected domains in the plane. Using these, we rigorously construct a good approximate solution to the problem.
We consider the nonlocal Hénon-Gelfand-Liouville problem $$ (-Δ)^s u = |x|^a e^u\quad\mathrm{in}\quad \mathbb R^n, $$ for every $s\in(0,1)$, $a>0$ and $n>2s$. We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when $$\dfrac{Γ(\frac n2)Γ(s)}{Γ(\frac{n-2s}{2})}\left(s+\frac a2\right)> \dfrac{Γ^2(\frac{n+2s}{4})}{Γ^2(\frac{n-2s}{4})}.$$
We classify finite Morse index solutions of the following Gelfand-Liouville equation \begin{equation*} (-Δ)^{s} u= e^u \ \ \text{in} \ \ \mathbb{R}^n, \end{equation*} for $1<s<2$ and $s=2$ via a novel monotonicity formula and technical blow-down analysis. We show that the above equation does not admit any finite Morse index solution with $(-Δ)^{s/2} u$ vanishes at infinity provided $n>2s$ and \begin{equation*} \label{1.condition} \frac{ Γ(\frac{n+2s}{4})^2 }{ Γ(\frac{n-2s}{4})^2} < \frac{Γ(\frac{n}{2}) Γ(1+s)}{ Γ(\frac{n-2s}{2})}, \end{equation*} where $Γ$ is the classical Gamma function. The …
Using Gelfand-Levitan theory on a half line, we construct a vessel for the class of potentials, whose spectral functions satisfy a certain regularity assumption. When the singular part of the spectral measure is absent, we construct a canonical model of the vessel. Finally, evolving the constructed vessel, we solve the Korteweg de Vries equation on the half line, coinciding with the given potential for $t=0$. It is shown that the initial value for x=0 is prescribed by this construction, but can be perturbed using an "orthogonal" to the problem measure. The results, presented in this work 1. in…
We construct stable solutions of $Δu + λe^u=0$ with Dirichlet boundary conditions in small tubular domains (i.e. geodesic $\varepsilon$--neighbourhoods of a curve $Λ$ embedded in $\mathbb{R}^n$), adapting the arguments of Pacard-Pacella-Sciunzi. We also show unicity of these solutions, in particular, we show that the stable branch of the bifurcation diagram is similar to the well-known nose-shaped diagram of the standard Gelfand problem in the unit ball. In this work, $Λ$ can be replaced by any compact smooth manifold embedded in $\mathbb{R}^n$.
Gelfand-Type problems in Random Walk Spaces
本文在随机游走空间框架下研究Gelfand型问题，证明了极值参数的存在性及对应解的存在性、稳定性与唯一性，并以加权图示例说明相关情形，拓展了该类问题的适用范围。
We consider the Lane-Emden system $-Δu = v^p$, $-Δv= u^θ$ in $\mathbb{R}^N$, and we prove the nonexistence of smooth positive solutions which are stable outside a compact set, for any $p, θ> 0$ under the Sobolev hyperbola.
We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-Δ)^su=e^u\quad\mathrm{in}\quad Ω\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$
In this paper we prove the Liouville type theorem for stable at infinity solutions of the following equation $$Δ_{m}^{3}u =|u|^{θ-1}u\;\;\; \mbox{in}\,\, \mathbb{R}^N,$$ for $1<m-1<θ<θ_{s, m}:=\frac{N(m-1)+3m }{N-3m}.$ Here $θ_{s, m}$ is a the classic critical exponent for $m-$ bi-harmonic equation.
Liouville-type results for some quasilinear anisotropic elliptic equations
本文证明了拟线性各向异性椭圆方程稳定解的Liouville型定理，并涵盖纯Finsler p-Laplacian情形。
We examine the weighted elliptic system \begin{equation*} \begin{cases} -Δu=(1+|x|^2)^{\fracα{2}} v,\\ -Δv=(1+|x|^2)^{\fracα{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$, $p>1$ and $α>0$. We prove Liouville type results for the classical positive (nonnegative) stable solutions in dimension $N<\ell+\dfrac{α(\ell-2)}{2}$ ($N <\ell+\dfrac{α(\ell-2)(p+3)}{4(p+1)}$) and $\ell \ge 5$, $p \in (1,p_*(\ell))$. In particular, for any $p>1$ and $α> 0$, we obtain the nonexistence of classical positive (nonnegative) stable solutions for any $N \le 12+5 α$ ($N\le 12+\…
In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -Δu=|x|^β v^{\vartheta},\\ -Δv=|x|^α |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ Ω, \end{equation*}where $Ω$ is a subset of $\mathbb{R}^N$, $N \ge 5$, $α>-4$, $0 \le β\le \dfrac{N-4}{2}$, $p>1$ and $\vartheta=1$. We first apply Pohozaev identity to construct a monotonicity formula and find their certain equivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicity formula} of solutions together with a {\it blowing down} sequence, we prove Liouville-type theorems of stable solutions …
In this paper we study solutions, possibly unbounded and sign-changing, of the following problem: -\D_λ u=|x|_λ^a |u|^{p-1}u, in R^n,\;n\geq 1,\; p>1, and a \geq 0, where \D_λ is a strongly degenerate elliptic operator, the functions λ=(λ_1, ..., λ_k) : R^n \rightarrow R^k, satisfies some certain conditions, and |.|_λ the homogeneous norm associated to the \D_λ-Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of R^n. First, we establish the standard integralestimates via stability property to deri…
We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ Ω\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)),$$ where $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. For bounded domain $Ω$ and for a stable weak solution of the above equation, we prove that the H…
We examine the general weighted Lane-Emden system \begin{align*} -Δu = ρ(x)v^p,\quad -Δv= ρ(x)u^θ, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\leqθ$ and $ρ: \mathbb{R}^N\rightarrow \mathbb{R}$ is a radial continuous function satisfying $ρ(x)\geq A(1+|x|^2)^{\fracα{2}}$ in $\mathbb{R}^N$ for some $α\geq 0$ and $A>0$. We prove some Liouville type results for stable solution and improve the previous works \cite{co, Fa, HU}. In particular, we establish a new comparison property (see Proposition 1.1 below) which is crucial to handle the case $1 < p \leq \frac{4}{3}$. Our resul…
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。
We examine the elliptic system given by {equation} \label{system_abstract} -Δu = v^p, \qquad -Δv = u^θ, \qquad \{in} \IR^N, {equation} for $ 1 < p \le θ$ and the fourth order scalar equation {equation} \label{fourth_abstract} Δ^2 u = u^θ, \qquad \{in $ \IR^N$,} {equation} where $ 1 < θ$. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (\ref{system_abstract}) (resp. (\ref{fourth_abstract})) provided $ N \le 10$ and $ 2 \le p \le θ$ (resp. $ N \le 10$ and $1 < θ$). Results for higher dimensions are also obtain…
In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-Δ)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable solutions provided that $N<10s$. This result is optimal when $s\uparrow 1$.
We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any $C^2(\mathbb{R}^N)$ positive entire semi-stable solution of the following Lane-Emden system…
We examine the degenerate elliptic system $$-Δ_{s} u = v^p, \quad -Δ_{s} v= u^θ, \quad u,v>0 \quad\mbox{in }\; \mathbb{R}^N=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where }\;\;\;\; s \geq 0\;\; \mbox{and} \;\;p,θ>0.$$ We prove that the system has no smooth stable solution provided $p,θ>0$ and $N_s< 2 + α+ β,$ where $$α= \frac{2(p+1)}{pθ- 1} \quad\mbox{and} \quad β= \frac{2(θ+1)}{pθ- 1}.$$ This result is an extension of some result in \cite{ MY}. In particular, we establish a new the integral estimate for $u$ and $v$ \;(see Proposition 1.1), which is crucial to deal with the case $0 …
Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System
本文研究Hénon-Lane-Emden系统稳定解的Liouville型定理，证明了在特定条件下猜想成立，并改进了相关文献结果。
Liouville theorems for stable solutions to the mean curvature equation
该研究针对平均曲率方程，在三类不同稳定条件的解中建立刘维尔型定理，将半线性Lane-Emden方程的相关结果推广至平均曲率方程。
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.
We prove that 0 the only classical solution of the Lane-Emden equation in the half-space which is stable outside a compact set. We also consider weak solutions and the case of general cones.
We develop a monotonicity formula for solutions of the fractional Toda system $$ (-Δ)^s f_α= e^{-(f_{α+1}-f_α)} - e^{-(f_α-f_{α-1})} \quad \text{in} \ \ \mathbb R^n,$$ when $0<s<1$, $α=1,\cdots,Q$, $f_0=-\infty$, $f_{Q+1}=\infty$ and $Q \ge2$ is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when $n > 2s$ and $$ \dfrac{Γ(\frac{n}{2})Γ(1+s)}{Γ(\frac{n-2s}{2})} \fr…
We prove a Liouville-type theorem for bounded stable solutions $v \in C^2(\R^n)$ of elliptic equations of the type (-Δ)^s v= f(v)\qquad {in $\R^n$,} where $s \in (0,1)$ {and $f$ is any nonnegative function}. The operator $(-Δ)^s$ stands for the fractional Laplacian, a pseudo-differential operator of symbol $|ξ|^{2s}$.
In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system \begin{align*} (-Δ)^s u = h(x)v^p,\quad (-Δ)^s v= h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} where $1<q\leq p$ and $h$ is a positive continuous function in $\mathbb{R}^N$ satisfying $\displaystyle{\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0$ with $\ell > 0.$ Our results generalize the results established in \cite{HHM16} for the Laplacian case (correspond to $s=1$) and improve the previous work \cite{TuanHoang21}. As a consequence, we prove classification…
Liouville Theorem for Lane Emden Equation of Baouendi Grushin operators
本文证明了对 Baouendi Grushin 算子的 Lane Emden 方程，当 p 小于 Joseph-Lundgren 指数且不同于 Sobolev 指数时，紧集外稳定的解只有零解，扩展了 Farina 的结果。
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。
We prove a Liouville property for uniformly almost localized (up to translations) H 1-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure. More precisely, we show that such solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H 1-functions with a momentum density that belongs to M + (R). Finally, we also get an asymptotic stability result for train of peakons.
In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation {equation*} Δ^2 u=|x|^a |u|^{p-1}u\;\ {in}\;\ Ω, {equation*}where $a \ge 0$, $p>1$ and $Ω\subset \mathbb{R}^n$ is an unbounded domain of $\mathbb{R}^n$, $n \ge 5$. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the {\it Pohozaev-type identity}, {\it monotonicity formula} of solutions and a {\it blowing down} sequence, which is used …
In this paper, some elliptic equation in a bounded open domain in $\mathbb{R}^N$ ($N\geq 2$) with $C^2$ boundary $\partialΩ$ is considered. The problem is driven by the regional fractional Laplacian, the infinitesimal generator of the censored symmetric $2α$-stable process in $Ω$. Probability theory asserts that the censored $2α$-stable process can not approach the boundary when $α\in(0,\frac12]$. For $α\in (0,\frac12]$, our purpose in this article is to show that non-existence of solutions bounded from above or bounded from below for the particular Poisson problem $$ (-Δ)^α_Ωu= 1 \quad {\rm i…
The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.
Inner regularity and Liouville theorems for stable solutions to the mean curvature equation
本文研究均曲率方程稳定解的内正则性和利乌维勒定理，证明了稳定解的正则性及在特定增长条件下的全局刚性，揭示了低维中稳定径向解的刚性特性。
In this paper, we establish Liouville type theorems for stable solutions on the whole space $\mathbb R^N$ to the fractional elliptic equation $$(-Δ)^su=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a classification of stable solutions to the fractional Lane-Emden system $$\begin{cases} (-Δ)^s u = v^p \mbox{ in }\mathbb R^N (-Δ)^s v = u^q \mbox{ in }\mathbb R^N \end{cases}$$ with $p>1$ and $ q>1$. In our knowledge, this is the first classification result for stable solutions of the fractional Lane-Emden system in literature.
In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-Δ_p u= f(x) F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in C^1(\mathbb{R})$. We refer to $f(x)$ as {\it weight} and to $F(u)$ as {\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0<C\le f$ then it does not have much impact on the stable solutions, however, a nonnegative weight that is $0\le f$ will push certain critical dimensions. This analytical ob…
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue …
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。
We study stable solutions of the following nonlinear system $$ -Δu = H(u) \quad \text{in} \ \ Ω$$ where $u:\mathbb R^n\to \mathbb R^m$, $H:\mathbb R^m\to \mathbb R^m$ and $Ω$ is a domain in $\mathbb R^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $Ω=\mathbb R^n$, and regularity results, when $Ω=B_1$, for stable solutions of the above system for a general nonlinearity $H \in C^1(\mathbb R ^m)$. Moreover, we pr…
Liouville type theorems for stable solutions of the weighted system involving the Grushin operator with negative exponents
本文研究含Grushin算子和负指数的一般加权系统，通过构造含最大根的条件证明其无稳定解，改进已有结果并推广到单个加权方程。
This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.
We investigate here the nonlinear elliptic Hénon type equation: $$\D^{2} u= |x|^a|u|^{p-1}u \; \,\,\mbox{in}\,\,\,\, \R^{n}_{+}, \quad \quad u =\frac{\partial u}{\partial x_n} = 0 \quad \mbox{in}\,\,\,\, \partial \R^{n}_{+},$$ with $p>1$ and $n\geq 2$. In particular, we prove some Liouville type theorems for stable at infinity solutions. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.
We study the quasilinear Dirichlet boundary problem \begin{equation}\nonumber \left\{ \begin{aligned} -Qu&=λe^{u} \quad \mbox{in}\quadΩ\\ u&=0 \quad \mbox{on}\quad\partialΩ,\\ \end{aligned} \right. \end{equation} where $λ>0$ is a parameter, $Ω\subset\mathbb{R}^{N}$ with $N\geq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)). $$ Here, $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $…
On Dancer's conjecture for stable solutions with sign-changing nonlinearity
本文在变号非线性项且无一维非常值稳定解及零点非退化条件下，证明了任意维数中一类二阶半线性椭圆方程的稳定解必为常数，部分回答了Dancer猜想。
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。
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$. This result, that was only known to be true for $n\leq4$, is optimal: $\log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $n\geq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result fo…
We prove $m$-dimensional symmetry results, that we call $m$-Liouville theorems, for stable and monotone solutions of the following nonuniformly elliptic equation \begin{eqnarray*}\label{mainequ} - div(γ(\mathbf x') \nabla u(\mathbf x)) =λ(\mathbf x' ) f(u(\mathbf x)) \ \ \text{for}\ \ \mathbf x=(\mathbf x',\mathbf x'')\in\mathbf{R}^d\times\mathbf{R}^{s}=\mathbf{R}^n, \end{eqnarray*} where $0\le m<n$ and $0<λ,γ$ are smooth functions and $f\in C^1(\mathbf R)$. The interesting fact is that the decay assumptions on the weight function $γ(\mathbf x') $ play the fundamental role in deriving $m$-Liou…
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.
Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity
本文研究了三维定常MHD和Hall-MHD系统在无穷远处特定渐近条件下的Liouville型定理，通过Lizorkin理论和Kato稳定性定理处理Hall项难点，证明了速度场和磁场在特定条件下均为常向量。
Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation
本文证明$p$-Laplace Lane--Emden方程在超临界范围内无径向对称假设的Liouville定理，给出尖锐阈值并解决Aubin--Talenti极值的Morse指标问题。
Regularity theory for mixed local-nonlocal problem involving general stable operators
本文针对含一般稳定Lévy型算子的混合局部-非局部椭圆方程，通过最大值原理和Liouville型结果，在适当假设下建立了解的内部及边界正则性。
We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional $p-$Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions.…
We are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: \begin{displaymath} Δu+\dfracμ{|x|^2}u+|x|^l |u|^{p-1}u=0 \qquad \textrm{in}\ \ Ω, \end{displaymath} where $Ω=\RN$, $\RN\setminus\{0\}$ for $N\geq3$, $p>1$, $l>-2$ and $μ<(N-2)^2/4$. Our results depend crucially on a new critical exponent $p=p_c(l,μ)$ and the parameter $μ$ in Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for $1<p<p_c(l,μ)$. We also observe a range of the expone…
Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
该研究针对具变号非线性项的半线性椭圆方程稳定解，在不同维数下建立了内部正则性估计，解决了相关公开问题并推广了已有结果，还证明了对应Liouville型定理。
This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to a…
We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb{M}$ without boundary, \begin{equation*} -Δ_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when $Δ_g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb R$ and $H_i\in C^1(\mathbb R^m) $ for $1\le i\le m$. This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is $\mathbb H(u)=(\partial_j H_i(u))_{i,j=1}^m$, is symmetric. We prove a stability inequality and a Poincaré type inequality …
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