Heisenberg群上非线性方程的存在性结果

本文通过变分方法给出了Ｈｅｉｓｅｎｂｅｒｇ 群上非线性方程的存在性结果     

Existence Results for Nonlinear Subelliptic Equations on the Heisenberg Group

§ 1.Introduction　 The aim of this paper is to establish existence results,by monotone method,for theproblemΔHnu + f((z,t) ,u) =0　　 in D,u| D =0 (1 .1 )where D is an open subset of the Heisenberg group HnandΔHn is the subelliptic Lapla-cian on Hn.We recall that Hnis the Lie group whose underlying manifold is Cn× R,n∈N,endowed with the group law(z,t) (z′,t′) =(z + z′,t+ t′+ 2 Imz .z′) ,(1 .2 )where for z,z′∈ Cnwe have letz .z′= nj=1zjz′j.Set zj=xj+ iyj.Then (x1 ,… ,xn…     

Existence Results for the Positive Solutions of a Fourth Order Nonlinear Equations on the Heisenberg Group

In this paper, we give some existence results for a fourth order nonlinear subelliptic equations on the Heisenberg group by the Leray-Schauder degree.     

Sem ilinear Equationswith CriticalExponentsonthe Heisenberg Group

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The Existence of Global Solution for a Class of Semilinear Equations on Heisenberg Group

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 Z2 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-ΔHnu = |u|p-1u,x∈Hnhas infinite positive solutions. Our approach can also be applied to study more general nonlinear problems.     

Heisenberg群上非线性方程的特征值问题

本文研究Heisenberg群上非线性特征值问题,利用Lerny－Schauder度给出一些存在性结果及解集的性质．     

Existence Results for Some Partial Integro-Differential Equations

In this note, we deal with semilinear integro-differential equations subject to homogeneous Dirichlet boundary conditions given on the boundaries of the sections. Even if the differentiation will be taken only in some directions, it is not possible to see the main problem parameterized by the other coordinates because of the non-local terms which also obliged the problem to be degenerate. We establish the existence of solutions by employing the singular perturbations method as a natural tool. The perturbed problems are classical, non-local, semilinear elliptic problems and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. Some improvement, concerning the existence of the solutions and the convergence results depending on the weights, will be established. The paper also gives an idea about the study of the anisotropic singular perturbations in the framework of weighted spaces.     

Regularity of Inhomogeneous Quasi-Linear Equations on the Heisenberg Group

We establish Hölder continuity of the horizontal gradient of weak solutions to quasi-linear $p$-Laplacian type non-homogeneous equations in the Heisenberg Group.     

A Critical Phenomenon for Sublinear Elliptic Equations in Cone-Like Domains

The authors of this paper study positive supersolutions to theelliptic equation -u = c|x|^–su^p in Cone-like domains ofR^N (N 2), where p, s R and c > 0. They prove that in thesublinear case p < 1 there exists a critical exponent p_*> 1 such that the equation has a positive supersolution ifand only if – < p < p_*. The value of p_* is determinedexplicitly by s and the geometry of the cone. 2000 MathematicsSubject Classification 35J60 (primary), 35B05, 35R45 (secondary).     

Comparison Principles for Some Fully Nonlinear Sub-Elliptic Equations on the Heisenberg Group

In this paper,we prove a form of the strong comparison principle for a class of fully nonlinear subelliptic operators of the form■H^2,sψ+L(·,ψ,■Hψ)on the Heisenberg group,which include the CR invariant operators.     

Existence of Random Attractors for 2D-Stochastic Nonclassical Diffusion Equations on Unbounded Domains

In this article, we prove the existence of a random attractor for stochastic nonclassical diffusion equations on unbounded domains, and the asymptotic compactness of the random dynamical system is established by a tail-estimates method.     

一类奇异临界椭圆方程非平凡解的存在性

研究了一类带Sobolev-Hardy临界指数的奇异椭圆方程,应用变分方法,通过能量估计和证明对应的能量泛函满足(PS)_c条件,运用山路引理得到了这类方程非平凡解的存在性.     

Heisenberg群上次椭圆p-Laplace方程的边界估计

研究次椭圆p-Laplace方程(P＞1)解的边界性质,通过建立Heisenberg群上带有区域内点到边界Carnot-Carath閛dory距离函数的Hardy型不等式,给出了有界域上次椭圆p-Laplace方程以及带非平凡位势的次椭圆p-Laplace方程的解在边界附近的若干估计.     

Existence Results for Quasilinear Elliptic Equations with Discontinuous Nonlinearities

The nonsmooth critical point theory is applied to prove the existence of solutions and multiple solutions of a quasilinear elliptic equation with discontinuous nonlinearities.     

Existence Results for Singular Boundary Value Problems on Unbounded Domains in Banach Spaces

In this paper, the Krasnosels’kii fixed point theorem in cones for strict set-contractions is used to investigate the existence of single and twin positive solutions for a class of a two-point boundary value problem of second-order nonlinear differential equations posed on an infinite interval. The nonlinearity, which may have a time-singularity, takes values in a general Banach space and has at most polynomial growth with respect to the unknown.     

Existence Results for Super-Liouville Equations on the Sphere via Bifurcation Theory

We are concerned with super-Liouville equations on $\mathbb{S}^2$, which have variational structure with a strongly-indefinite functional. We prove the existence of nontrivial solutions by combining the use of Nehari manifolds, balancing conditions and bifurcation theory.     

Heisenberg群上半线性Kohn-Laplace′s方程的Dirichlet问题

本文用上下解方法,获得半线性次椭圆方程Dirichlet问题的-些存在性结果.     

Heisenberg群上半线性Kohn-Laplace's方程的Dirichlet问题(英文)

本文用上下解方法,获得半线性次椭圆方程Dirichlet问题的一些存在性结果.     

Heisenberg群上半线性Kohn—Laplacés方程的Dirichlet问题

本文用上下解方法，获得半线性次椭圆方程Dirichlet问题的一些存在性结果．     

Heisenberg群上抛物型方程解的平均值公式

证明了Heisenberg群上一类含次p-Laplace抛物型方程粘性解的等价定理,在这个等价定理的基础上证明一个函数为该方程的解当且仅当该函数在粘性意义下满足渐近平均值公式.