Geyang Du, Shulin Zhou
We first give some apriori estimates of positive radial solutions of $p$-Laplace Hénon equation. Then we study the local and global properties of those solutions. Finally, we generalize some radial results to the nonradial case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Degeneracy Set Comparison Principle and Free Boundary Estimates for Hénon-type Infinity Laplace equations
本文针对带有退化权重等项的Hénon型无穷拉普拉斯方程，证明了退化集比较原理，建立了自由边界点的正则性与非退化估计，明确了解脱离零相位的速率。
Well-posedness and global existence for Hardy--Hénon parabolic equations in Herz spaces
研究Hardy-Hénon抛物方程，在参数\(\gamma\)适当假设下，通过建立温和解的局部和整体适定性，得出齐次Herz空间中的相关结果。
We prove the existence of nonradial solutions for the Hénon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $α$. For sign-changing solutions, the case $α=0$ -- Lane-Emden equation -- is included. The obtained solutions form global continua which branch off from the curve of radial solutions $p\mapsto u_p$, and the number of branching points increases with both the number of nodal zones and the exponent $α$. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solut…
We consider the Hardy-Hénon parabolic equation $u_t-Δu =|x|^a |u|^{p-1}u$ with $p>1$ and $a\in {\mathbb R}$. We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.
Asymptotically self-similar global solutions for Hardy-Hénon parabolic equations
该研究针对Hardy-Hénon抛物型方程，为加权Lorentz空间中的一大类初始数据构造了渐近自相似整体解，揭示了其依初始数据衰减速度的两种渐近类型，统一了不同γ情形并发现复值初始数据的新渐近行为，还证明了渐近轮廓稳定性与超临界初始数据无局部正解。
This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\RR^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\RR^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time…
Blow-up and global mild solutions for a Hardy-Hénon parabolic equation on the Heisenberg group
研究Heisenberg群上Hardy-Hénon抛物方程，确定了Fujita临界指数并证明在临界指数附近解的爆破与全局存在性，推广了已知结果。
Non-radial solutions for the critical quasi-linear Hénon equation involving $p$-Laplacian in $\R^N$
本文研究 $\R^N$ 中涉及 $p$-Laplacian 的临界拟线性 Hénon 方程，通过线性化分析、逼近方法和分岔理论证明在参数 $\alpha$ 取特定临界值时存在非径向解，将 Laplace 算子的经典结果推广到一般 $p$-Laplace 算子情形。
Fujita Phenomenon for a Mixed Local-Nonlocal Hardy-Hénon Equation with Regularly Varying Time Weights
研究了一类混合局部-非局部扩散算子驱动的半线性抛物型方程的Cauchy问题，探讨了临界Fujita指数下的有限时间爆破和全局存在性条件，并首次分析了非线性扩散方程在具有此类时间依赖系数下的爆破现象。
On the Hardy-Hénon heat equation with an inverse square potential
研究带有反平方势的Hardy-Hénon抛物方程的Cauchy问题，建立热半群在加权Lebesgue空间中的精确衰减估计，并讨论局部解的存在性和临界空间中的全局解的条件。
Global self-similar solutions for Hardy-Hénon equations with linear and quasilinear diffusion
研究Hardy-Hénon方程中具有线性和拟线性扩散的全局自相似解的分类及存在性，揭示了临界指数对解结构的影响。
We prove global comparison results for the $p$-Laplacian on a $p$-parabolic manifold. These involve both real-valued and vector-valued maps with finite $p$-energy.
In this paper, we establish higher integrability of the gradient of the solution of the quasilinear elliptic equation $Δ_Au=\text{div}\left(\frac{a(|F|)}{|F|}F\right)$ in $\mathbb{R}^n$, where $Δ_Au$ is the so called A-Laplace operator.
Global compactness results for fractional $p$-Laplace Hardy Sobolev operator on a bounded domain
本文研究了分数 $p$-拉普拉斯 Hardy-Sobolev 运算符在有界域上的非线性临界问题，建立了全局紧致性结果。
Local and global $C^{1,β}$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type
本文研究了均匀椭圆$p$-拉普拉斯和Orlicz-拉普拉斯类型quasilinear方程的局部和全局$C^{1,β}$正则性，证明了在内部和边界上的梯度Hölder连续性。
Global UCP For Parabolic Fractional $p$-Laplace Equation With Very Rough Potentials
研究抛物分数$p$-拉普拉斯方程在非常粗糙势函数下的全局唯一持续原理，提出新方法避免扩展技术和Carleman估计，解决局部问题仍存开放性。
A parabolic Hardy-Hénon equation with quasilinear degenerate diffusion
本文研究了含拟线性退化扩散和奇异势的抛物型Hardy-Hénon方程的适定性与爆破现象，利用Caffarelli-Kohn-Nirenberg不等式揭示了奇异势对有限时间爆破的抑制效应，并确立了不同指数区间下解的局部存在性、唯一性、全局存在性及爆破条件。
Global minimizers of the two-phase Bernoulli problem with the $p$-Laplace operator
研究两相 $p$-Laplace Bernoulli 问题的 Lipschitz 全局解的分类，证明在正则两相点附近具有 $C^{1,\eta}$ 正则性。
We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $ -Δu + F(u) = f $ with Dirichlet or Neumann boundary conditions on a precompact domain $ Ω\subset \mathbb{R}^{n}$, where $ F : \mathbb{C} \rightarrow \mathbb{C} $ is Lipschitz. The same method gives a solution to $ - Δ_{g} u + F(u) = f $ for these boundary conditions on a smooth, compact Riemannian manifold $ (M, g) $ with $ \mathcal{C}^{1} $ boundary, where $ - Δ_{g} $ is the Laplace-Beltrami operator. We also apply parametrix methods to discuss an integral version …
We study existence and regularity properties of solutions to the singular $p$-Laplacean parabolic system in a bounded domain $Ω$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(Ω))$, $\varepsilon\geq0$, integrability properties of the second spatial derivatives and of the time derivative of the solutions. Hence, for suitable $p$ and exponents $r,\,q$, by Sobolev embedding theorems, we deduce global regularity of $u$ and $\nabla u$ in Hölder spaces. Finally we prove a global pointwise bound for the solution under the assumption $p>\frac{2n}{n+2}$.
An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the $p$-Laplace equation and system, with right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $\mathrm{BMO}$ and $\mathrm{VMO}$ spaces…
$C^{1, α}$-regularity for functions in solution classes and its application to parabolic normalized $p$-Laplace equations
本文在椭圆性常数足够接近时证明解类函数的全局$C^{1, α}$正则性，并借助紧性论证与迭代过程，将其应用于$p$接近2时的抛物型规范化$p$-Laplace方程，得到全局正则性结果。
The Neumann problem for the generalized Hénon equation. Local analysis
研究单位球内广义Henon方程Neumann边界值问题正径向解的性质，证明在特定条件下能量函数的局部极小性。
We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-Δ_p u=μ$ with $p\in (1,2)$. The cases when $p\in \big(2-\frac 1 n,2\big)$ and $p\in \big(\frac{3n-2}{2n-1},2-\frac{1}{n}\big]$ were studied in [9] and [22], respectively. In this paper, we improve the results in [22] and address the open case when $p\in \big(1,\frac{3n-2}{2n-1}\big]$. Interior and global modulus of continuity estimates of the gradients of solutions are also established.
Boundary Hölder gradient estimates for parabolic $p$-Laplace type equations
本文研究了抛物 $p$-Laplace 型方程粘性解的边界正则性，获得了边界逐点和全局 $C^{1,α}$ 正则性。
A unified clasification of Liouville properties and nontrivial solution for fractional elliptic equations with general Hénon-type superquadratic and gradient growth
针对带一般Hénon型超二次及梯度增长的分数阶椭圆方程，建立统一分析框架确定临界平衡γ+p=2s，划分三类情形得到Liouville型结果、解的存在性等结论，证明依赖新非局部分析技术，贡献可与该方向近期深度研究媲美。
As the main problem, the bi-Laplace equation $Δ^2u=0 (Δ=D_x^2+D_y^2)$ in a bounded domain $Ω\subset \re^2$, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial Ω$ is considered. In addition, there is a finite collection of curves $$Γ= Γ_1\cup...\cupΓ_m \subset Ω, \quad \mbox{on which we assume homogeneous Dirichlet} \quad u=0,$$ focusing at the origin $0 \in Ω$ (the analysis would be similar for any other point). This makes the above elliptic problem overdetermined. Possible types of the behaviour of solution $u(x,y)$ at the tip $0$ of such admissible multip…
We consider an inverse problem of determining the coefficients of a fractional $p\,$-Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficents. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.
Weak Harnack estimates for a doubly nonlinear nonlocal p-Laplace equation
针对双重非线性非局部p-Laplace方程的弱上解，建立了带最优抛物尾的新型弱Harnack估计，利用正性扩张和测度理论，仅需局部正性，凸显非局部特征。
In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global ex…
We study the global solution curves, and prove the existence of infinitely many positive solutions for three classes of self-similar equations, with $p$-Laplace operator. In case $p=2$, these are well-known problems involving the Gelfand equation, the equation modeling electrostatic micro-electromechanical systems (MEMS), and a polynomial nonlinearity. We extend the classical results of D.D. Joseph and T.S. Lundgren \cite{JL} to the case $p \ne 2$, and we generalize the main result of Z. Guo and J. Wei \cite{GW} on the equation modeling MEMS.
In this paper we deal with an inhomogeneous parabolic Dirichlet problem involving the 1-Laplacian operator. We show the existence of a unique solution when data belong to $L^1(0,T;L^2(Ω))$ for every $T>0$. As a consequence, global existence and uniqueness for data in $L^1_{loc}(0,+\infty;L^2(Ω))$ is obtained. Our analysis retrieves the results of \cite{SW} in a correct and complete way.
Regularity and Separation for $p$-Laplace operators
本文研究退化椭圆系数下的 $p$-Laplace 及 Grušin 型算子，揭示弱解的分离与跳跃间断，并证明抛物方程 mild 解的全局 $L^1$-$L^\infty$ 正则化及衰减估计。
Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity
本文利用 Bernstein 技巧结合近似方法，证明了抛物 $p$-Laplace 方程解的时间导数有界性，获得了最优正则性结果并关联 $C^{p'}$-猜想，同时将方法推广至完全非线性和拟线性退化抛物问题的全局正则性。
Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in $L^2$. The nonlinear expression of the gradient under the divergence operator is shown to belong to $W^{1,2}$, and hence to enjoy the best possible degree of regularity. Moreover, its norm in $W^{1,2}$ is proved to be equivalent to the norm of the right-hand side in $L^2$. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local esti…
Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms
本文研究含奇异低阶项的1-Laplace齐次Dirichlet问题，在f可和性与h无穷远处行为的精确条件下证明BV全局有限能量解的存在性，并推广了Lazer-McKenna经典结果。
Global solvability of the Laplace equation in weighted Sobolev spaces
本文研究了拉普拉斯方程在加权Sobolev空间中的弱解与强解的可解性，证明了弱解在特定条件下满足强解条件，并指出此类问题需特殊技术处理。
Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-Hénon Type Parabolic Equations
研究带Hardy-Hénon权重的分数阶半线性抛物方程解的寿命，建立了与Fujita临界指数相关的尖锐上下界，并推广了经典结果。
We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p,q)$-Laplace equation $-Δ_p u -Δ_q u = α|u|^{p-2}u + β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $α, β\in \mathbb{R}$. A curve on the $(α,β)$-plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the $p$- and $q$-Laplacians under zero Dirichlet boundary condition are linear…
Global branching for semilinear fractional Laplace with sublinear nonlinearity
本文研究次线性分数阶椭圆问题$(P_λ^s)$正解的存在性、不存在性和多重性，通过先验估计、上下解方法及经典连接定理证明至少存在两个不同正弱解，扩展了已有结果至非局部情形。
Calderón-Zygmund gradient estimates for $p$-Laplace systems with BMO complex coefficients
本文研究了复系数下$p$-Laplace系统的Calderón-Zygmund梯度估计，通过弱解的存在性和唯一性条件，推导出全局梯度界和Morrey空间正则性。
Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces
本文针对测度空间上带自伴算子的半线性阻尼波动方程，在热半群衰减估计下建立推广的松村估计，证明小数据全局存在性，并涵盖多种重要算子实例。
The $m$-Laplace equation with a gradient term II: classification in the second critical case
本文分类了带梯度项的 $m$-Laplace 方程在第二临界情形下的正弱解，证明解为常数或显式径向解族，无需全局有界或能量假设。
A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calderón-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calderón-Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical $(-Δ_p)^1$-Laplace …
We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.
Gradient continuity estimates for elliptic equations of singular $p$-Laplace type with measure data
本文在系数满足L²平均Dini条件下，通过新的比较估计建立了具测度数据的p-Laplace型椭圆方程梯度的内部与全局逐点估计，并推广了系数Dini连续情形下的已有结果。
Gradient Hölder regularity for singular fractional $p$-Laplace equations
该研究针对奇异分数阶p-拉普拉斯方程，通过内在过剩衰减论证等方法，证明了特定参数范围内有界弱解的梯度Hölder正则性，为相关未解决问题提供了部分解答。
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。
Second order regularity for degenerate p-Laplace type equations with log-concave weights
该研究针对带可能在边界退化的对数凹权重的凸区域p-拉普拉斯型方程，分别给出有界区域的精确全局二阶估计与无界区域的边界局部估计，填补了p=2情形下的相关研究空白。
Global second order optimal regularity for the vectorial $p$-Laplacian
针对向量值 $p$-Laplace 方程，本文获得了解的最优正则性结果，并解决了应力场的全局二阶估计问题。
We prove in this paper the global Lorentz estimate in term of fractional-maximal function for gradient of weak solutions to a class of p-Laplace elliptic equations containing a non-negative Schrödinger potential which belongs to reverse Hölder classes. In particular, this class of p-Laplace operator includes both degenerate and non-degenerate cases. The interesting idea is to use an efficient approach based on the level-set inequality related to the distribution function in harmonic analysis.
The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type \begin{align*} \begin{cases} \mathrm{div}(|\nabla u|^{p-2}\nabla u) &= f+ \ \mathrm{div}(|\mathbf{F}|^{p-2}\mathbf{F}) \qquad \text{in} \ \ Ω, \\ \hspace{1.2cm} u &=\ g \hspace{3.1cm} \text{on} \ \ \partial Ω, \end{cases} \end{align*} in Lorentz space, with given data $\mathbf{F} \in L^p(Ω;\mathbb{R}^n)$, $f \in L^{\frac{p}{p-1}}(Ω)$, $g \in W^{1,p}(Ω)$ for $p>1$ and $Ω\subset \ma…
A PDE Framework of Consensus-Based Optimization for Objectives with Multiple Global Minimizers
本文提出一种简单的CBO变体以解决具有多个全局极小值的非凸优化问题，证明相应非线性Fokker--Planck方程解的存在性及向多分量极小值集合的指数集中，并表明对应的Euler-Maruyama离散化算法趋于收敛到多个全局极小值。
Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth
本文针对p-Laplace型各向异性椭圆算子的Dirichlet和Neumann边值问题，在最弱右端项假设下建立了解的全局Lipschitz正则性，结果适用于凸域或边界曲率满足最小可积性假设的域。
We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hoelder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Lad…
Stability and maximal regularity of the diffusion equation on unbounded surfaces
本文研究无界曲面上扩散方程的稳定性与极大正则性，在无曲率假设下证明平缓斜率时存在全局强解并具$L^2$渐近稳定性，且在外力满足特定条件时推导极大正则性与Hölder连续性。
Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds
本文通过结合Bochner公式、Sobolev不等式和Nash-Moser方法，研究了带有梯度依赖反应的(p,q)-拉普拉斯方程的局部和全局行为，并得出了利奥维勒型定理。
We study the global-in-time Strichartz estimates for the Schrödinger equation on a class of scattering manifolds $X^{\circ}$. Let $\mathcal{L}_V=Δ_g+V$ where $Δ_g$ is the Beltrami-Laplace operator on the scattering manifold and $V$ is a real potential function on this setting. We first extend the global-in-time Strichartz estimate in Hassell-Zhang \cite{HZ} on the requirement of $V(z)=O(\langle z\rangle^{-3})$ to $O(\langle z\rangle^{-2})$ and secondly generalize the result to the scattering manifold with a mild trapped set as well as Bouclet-Mizutani\cite{BM} but with a potential. We also obt…
A comparison estimate for singular $p$-Laplace equations and its consequences
本文解决了强奇异情形下 $p$-Laplace 型椭圆方程缺乏比较估计的问题，在 $1<p<3/2$ 范围内证明了比较估计，并推导出逐点估计、Poincaré 型不等式及全局加权估计等推论。
Some results on cross-diffusion systems with entropy structure are reviewed. The focus is on local-in-time existence results for general systems with normally elliptic diffusion operators, due to Amann, and global-in-time existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion systems with an additional Laplace structure. The boundedness-by-entropy method allows for global bounded weak solutions to certain diffusion systems. Furthermore, a partial result on the uniqueness of weak solutions is recalled, and some open problems are presented.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。