On semilinear elliptic equations with global coupling
Shinji Kawano
考虑了一类含有整体耦合项的非线性椭圆型方程。我们对这个问题所有可能的正解的存在性进行分类。
We consider nonlinear elliptic equations which contains global coupling as a nonlinear term. We classify the existence of all possible positive solutions to this problem.
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In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
Semilinear elliptic equations for the fractional Laplacian with Hardy potential
本文研究一类含Hardy位势的半线性分数阶椭圆型方程非负分布解的存在性和不存在性。
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
Positive solutions of Robin problem for semilinear elliptic equations and a threshold result
本文证明了半线性抛物型方程Robin问题的一个阈值结果
n this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with perturbation arguments involving singular terms.
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.
Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential
本文研究了一类含平方反比势的半线性椭圆型方程的非负分布上解的不存在性。
Uniqueness of singular solution of semilinear elliptic equation
本文研究了一类椭圆型方程解在0附近的渐近性态。建立了奇异解的唯一性
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
Uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities
考虑具有双幂非线性项的半线性椭圆型方程。保证正解存在的条件是众所周知的。在本文中，我们注意到欧阳和施提出的保证唯一性的附加条件是不必要的。
We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of the solutions, under general assumptions on the nonlinearity.
We show an existence of a weak solution of a degenerate and/or singular semilinear elliptic boundary value (nonhomogeneous) problem lying between a given weak subsolution and a given weak supersolution. It has been applied for an existence result of large solution to a similar problem.
A remark on the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities
研究了具有双幂非线性项的半线性椭圆型方程正解的唯一性。我们从Peletier和Serrin的经典文章中的论证出发，推出了唯一性，从而恢复了欧阳和施的部分唯一性结果。
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
We consider a semilinear elliptic equation in a bounded domain with zero boundary conditions. The nonlinearity is discontinuous and monotone, but it is not a Carathéodory's function. The existence theorem has been proved.
In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.
Existence and uniqueness conditions of positive solutions to semilinear elliptic equations with double power nonlinearities
本文给出了具有双幂非线性项的半线性椭圆型方程正解存在唯一性的等价条件。作为奖励，我们给出了一个更简单的证明，我们以前的结果，唯一性条件来自存在性条件。
A uniqueness result for some singular semilinear elliptic equations
给定$Ω$是$\mathbb{R}^N$的有界开子集，考虑奇异半线性椭圆型方程$-Δ\,u\,=\,\frac{f}{u^β}$在$H^1_{loc}(Ω)$中的非负解，在零Dirichlet边界条件下.对于$β>0$和$f\in L^1(Ω)$，我们证明了解是唯一的。
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities
考虑有界光滑区域上奇异半线性椭圆型方程的正解，Dirichlet边界条件为零.我们通过移动平面方法给出了解的H^1_0部分的一些弱最大值原理和强最大值原理，从而可以推导出解的对称性和单调性。
Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions
给出了一类具有Laplace-Beltrami算子的线性和非线性高阶边值条件的Laplace-Beltrami算子半线性椭圆型边值问题可解的充要条件.
Nondegeneracy of Positive solutions of a Semilinear elliptic problem in the Hyperbolic space
本文研究N维双曲空间中半线性椭圆型方程有限能量正解的非退化性质。我们将证明，退化只发生在一个N维子空间。我们将证明在测地线球的情况下，正解是非退化的。
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.
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.
Almgren-type monotonicity methods for the classification of behavior at corners of solutions to semilinear elliptic equations
讨论了研究具有圆锥形边界点的区域中半线性椭圆型方程解在角点附近的渐进行为的单调性方法。对于足够接近直锥面的边界轮廓，排除了在渐近展开的首项中存在的边界。
Bifurcation results for semilinear elliptic problems in R^N
本文对R^N中的半线性椭圆问题，得到了从$-Δ$的谱底分叉的解族.这个问题是变分的性质，我们采用非线性减少方法，使我们能够搜索的解决方案作为临界点的合适的泛函定义在有限维流形。
This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.
We will present some rigidity results for solutions to semilinear elliptic equations of the form $\Deltau = W'(u)$, where W is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouvlle-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.
The purpose of this paper is to study the existence of solutions for semilinear elliptic system driven by fractional Laplacian and establish some new existence results which are obtained by virtue of the local linking theorem and the saddle point theorem. To make the nonlinear scheme feasible, rigorous analysis of the function space involved and corresponding energy functional is necessary.
Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem
定常涡的三维无粘不可压缩流的欧拉方程和浅水方程的构造和渐近趋于奇异涡丝。该构造是基于半线性椭圆问题解的渐近研究。
Boundary singularities for weak solutions of semilinear elliptic problems
构造了一类半线性椭圆型方程的正弱解，该方程在给定的光滑有界N维区域的边界上在适当的迹意义下为零，但在边界的给定孤立点处为奇.类似的结构进行的解决方案是奇异的任何给定的嵌入子流形的边界。
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.
Semilinear Elliptic Equations and Fixed Points
本文研究了有界区域$Ω\subset\mathbb{R}^N$中一类半线性椭圆型方程，$N\geq 3$，其边界为$C\sp{1,1}$.利用两个算子和的Krasnoselskii型不动点的一个新结果，证明了一个强解的存在性原理.我们给出两个例子，其中的非线性可以是关键的。
Homogenization of a semilinear elliptic problem
针对系数可能不连续的半线性椭圆方程，研究证明细尺度解的存在唯一性与先验估计，并通过双尺度收敛推导均匀化方程及校正子结果。
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
We consider the second order semilinear elliptic system $Δu= p\left( x\right) v^α,$ $Δv= q\left(x\right) u^β,$ where $x \in \mathbf{R}^N,$ $N \geq 3,$ $α$ and $β$ are positive constants, $p$ and $q$ are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions $p$ and $q$ are of slow decay.
Uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities, revised eddition
研究了具有双幂非线性项的半线性椭圆型方程正解的唯一性。保证这类方程正解存在的条件早已为人们所知。另一方面，为了唯一性，1998年Ouyang和Shi提出了相当技术性的附加条件。在本文中，我们注意到这个附加条件是不必要的。
We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set. We make no assumptions about the open set, very mild regularity assumptions on the semilinearity, plus a coerciveness assumption which depends on the optimal Poincare-Steklov constant. The proof is based on Schaefer's fixed point theorem applied to a sequence of truncated problems. We state a simple uniqueness result. We also generalize the results to Robin boundary conditions.
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 study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case.
On a semilinear elliptic systems in Hyperbolic space
本文考虑半线性椭圆型方程组\displaystyle -Δ_{ \mathbb{H}^{N} }u= |v| ^{p-1}v，\displaystyle -Δ_{ \mathbb{H}^{N} }v= |u| ^{q-1}u，在整个双曲空间$\mathbb{H}^{N}$中.我们建立了正解的衰减估计和对称性质。与欧几里得空间$\mathbb{R}^N$中的相应问题不同，我们证明了问题存在非负非平凡解.
We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $Ω\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(Ω)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.
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.
Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$
证明了半线性椭圆型方程$-Δu=f(u)$在整个$R^N$中不存在极限在无穷远的非常数全局极小解，其中$f\in C^1(R)$是一般非线性方程，$N\geq 1$是任意维数的方程.作为这个结果的推论，我们建立了非常数有界径向全局极小的前一个方程的不存在性。
We study the semilinear Poisson equation \begin{equation} \label{pro} Δu = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1. \end{equation} Our main results provide conditions on $f$ which ensure that weak solutions of this equation belong to $C^{1,1}(B_{1/2})$. In some configurations, the conditions are sharp.
Radial Symmetry of Large Solutions of Semilinear Elliptic Equations with Convection
研究了半线性椭圆型问题Δu + \nabla h的大解的径向对称性. \nabla u = f（|x|，u），我们提供了尖锐的条件下，该问题有一个径向的解决方案。这个结果与无穷远处解的增长率无关。
Energy estimates for a class of semilinear elliptic equations on half Euclidean balls
对于一类具有临界Sobolev指数和边界条件的半线性椭圆型方程，证明了爆破解的逐点估计和能量估计.这类方程的一个特例是局部定义的规定数量曲率和中曲率型方程。
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
Solutions without any symmetry for semilinear elliptic problems
本文证明了非线性Klein-Gordon或薛定谔方程$$ Δu-u+ u^3 =0 , $$在${\bf R}^2$中存在无穷多个能量有限且最大对称群约化为恒等的孤波.
Locating the peaks of semilinear elliptic systems
考虑一类弱耦合奇摄动半线性椭圆型方程组。首先，我们得到了相应的基态能量函数$Σ$的Lipschitz正则性结果以及左导数和右导数的表示公式。然后，我们证明了在次微分意义下，解的集中点位于$Σ$的临界点附近。
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
Nonexistence results for semilinear elliptic equations on weighted graphs
本文研究无限图上带位势的半线性椭圆不等式，在距离的 Laplacian 上界与加权球体积增长条件下，证明非负非平凡解不存在，并验证条件最优。
We establish a logarithmic stability inequality for the inverse problem of determining the non linear term, appearing in a semilinear BVP, from the corresponding Dirichlet-to-Neumann map (abbreviated to DtN map in the rest of this text). Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester [Commun. Pure Appl. Math. 47 (1994), 1403-1410].
In this paper we prove some symmetry results for entire solutions to the semilinear equation $-Δu=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give some sufficient conditions for them to be radially symmetric with respect to those variables, such as periodicity or the pointwise decay of some derivatives.
Semilinear elliptic inequalities in the exterior of a compact set
研究了半线性椭圆不等式$-Δu\geqφ(δ_K(x))f(u)$在$R^N\setminus K,$中的性质，其中$φ, f$是非负连续函数，$K\subset R^N$ $(N\geq 2)$是紧集，$δ_K(x)={\rm dist}(x,\partial K)$ .利用$φ$和$f$得到了$C^2$正解存在的最优条件.我们的分析强调了紧集$K$的几何所起的作用。
On the Fountain Theorem for Continuous Functionals and Its Application to a Semilinear Elliptic Problem in $\mathbb{R}^2$
本文基于弱斜率框架建立连续喷泉定理，推广Willem定理，并应用于半线性椭圆问题的研究。
In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -Δu=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), & \end{array} \right. $$ where $G$ is either $\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a continuous function assuming critical and subcritical values.
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