Existence and Asymptotic Behavior of Ground State Solutions for Quasilinear Schr?dinger Equations with Unbounded Potential

The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients. By applying the mountain pass theorem and the concentration compactness principle, they get a ground state solution. Moreover, the asymptotic behavior of the ground state solution is also obtained.     

EXISTENCE RESULTS FOR DEGENERATE ELLIPTIC EQUATIONS WITH CRITICAL CONE SOBOLEV EXPONENTS

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.     

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = (σ+m)n/(n-σ-2) is its critical exponent provided max{-1, [(1-m)n-2]/(n+1)} σ n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Furthermore, we demonstrate that if max{1, σ + m} p ≤ pc, then every positive solution of the equations blows up in finite time; whereas for ppc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n ≤σ+2.     

ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.     

SINGULAR PERTURBATION OF THREE-POINT BOUNDARY VALUE PROBLEM FOR THIRD ORDER DIFFERENTIAL EQUATIONS

In this paper, by the theory of differential inequalities, we study the existence and uniqueness of the solution to the three-point boundary value problem for third order differential equations. Furthermore we study the singular perturbation of three-point boundary value problem to third order quasilinear differential equations, construct the higher order asymptotic solution and get the error estimate of asymptotic solution and perturbed solution.     

On a nonlinear elliptic problem with critical potential in R~2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in R2. By establishing a weighted inequality with the best constant, determine the critical potential in R2, and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexis tence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

Solutions for a class of singular nonlinear boundary value problem involving critical exponent

In this paper, we consider the existence of multiple solutions for a class of singular nonlinear boundary value problem involving critical exponent in Weighted Sobolev Spaces. The existence of two solutions is established by using the Ekeland Variational Principle. Meanwhile, the uniqueness of positive solution for the same problem is also obtained under different assumptions.     

奇摄动问题中阶梯状解的缝接法

In view of singularly perturbed problems with complex inner layer phenomenon,including contrast structures(step-step solution and spike-type solution),corner layer behavior and right-hand side discontinuity,we carry out the process with sewing connection.The presented method of sewing connection for singularly perturbed equations is based on the two points singularly perturbed simple boundary problems.By means of sewing orbit smoothness,we get the uniformly valid solution in the whole interval.It is easy to prove the existence of solutions and deal with the high dimensional singularly perturbed problems.     

Energy Scattering Theory for Electromagnetic NLS in Dimension Two

We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schr?dinger equations with scaling critical magnetic potentials in dimension two.In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov–Bohm potential which decays likely the Coulomb potential |x|~(-1).     

奇异积分算子交换子在变指数空间上的特征

The main purpose of this paper is to characterize the Lipschitz space by the boundedness of commutators on Lebesgue spaces and Triebel-Lizorkin spaces with variable exponent.Based on this main purpose, we first characterize the Triebel-Lizorkin spaces with variable exponent by two families of operators. Immediately after, applying the characterizations of TriebelLizorkin space with variable exponent, we obtain that b ∈■β if and only if the commutator of Calderón-Zygmund singular integral operator is bounded, respectively, from■ to■,from■ to■ with■. Moreover, we prove that the commutator of Riesz potential operator also has corresponding results.     

On singular nonhomogeneous elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent

In this paper, we establish the existence of multiple solutions for nonhogeneous singular elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent, by using Ekeland’s Variational Principle and Mountain Pass Theorem without Palais Smale conditions.     

The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):{-△u αu=f(u),x∈Ω, x∈Ω,δu/δr=0,x∈δΩ,is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow.     

Solutions to nonlinear Schr?dinger equations with singular electromagnetic potential and critical exponent

We investigate existence and qualitative behavior of solutions to nonlinear Schr?dinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of degree –1, including the Aharonov–Bohm class. In particular, by variational arguments we prove a result of multiplicity of solutions distinguished by symmetry properties.     

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.     

Blow-up for Strauss type wave equation with damping and potential

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open.     

具有临界增长的拟线性椭圆型混合边值问题的非平凡解

本文给出了ＲＮ中一类有界区域εαＮ上临界增长拟线性椭圆型方程Ｆｉ（ｘ，ｕ，Ｄｕ）－Ｆｕ（ｘ，ｕ，Ｄｕ）＝０的混合边值问题非平凡解的存在性。     

一类奇异临界双调和椭圆方程的群不变解

讨论一类含有Hardy-Sobolev临界指数项的奇异双调和椭圆方程,应用Lions集中紧性原理、Palais对称临界原理、Hardy-Rellich型不等式和变分方法,证明了方程在适当条件下群不变解的存在性和多重性.     

Existence result for a class of singular quasilinear elliptic equations with critical growth

In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations with critical exponents. The proofs rely on a non-smooth critical point theory, and some techniques used by Brezis and Nirenberg.     

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p _* : =  p(N  −  1)/(N  −  p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.