Existence of solutions for the p-Laplacian with critical Sobolev exponent and convection

In this article, we study the existence of solutions for the p-Laplacian involving critical Sobolev exponent and convection based on the theory of the Leray–Schauder degree for non-compact mappings.     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .

     

带有临界Sobolev指数的奇异双调和问题

LetΩR~N be a smooth bounded domain such that 0∈Ω,N≥5,2~*:=(2N)/(N-4) is the critical Sobolev exponent,and f(x) is a given function.By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problemΔ~u-μu/(|x|~4)=|u|~(2~*-2)u f(x) with Dirichlet boundary condition on Ωunder some assumptions on f(x) andμ.     

On the existence of an extremal function in critical Sobolev trace embedding theorem

Sufficient conditions for the existence of extremal functions in the trace Sobolev inequality and the trace Sobolev-Poincaré inequality are established. It is shown that some of these conditions are sharp.     

Singular elliptic problems with lack of compactness

We consider the following nonlinear singular elliptic equation

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.

     

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohoaev-type identity.AMS Subject Classification: Primary: 35J65; Secondary: 35B33.     

涉及第一特征值和临界指数的一类椭圆方程

本文给出了半线性椭圆方程-△u=λ1u |u|^2 -2u τ(x，u)的Dirichet问题在对非线性次临界扰动项τ(x，u)增加适当条件后非平凡解的存在性定理等．     

On asymptotic stability in energy space of ground states of NLS in 2D

We transpose work by K. Yajima and by T. Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.     

On the “Destruction” of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin-Bona-Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin-Bona-Mahony type, we prove the “failure” of a “weakened” solution in finite time.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002     

On the H-property of functionals in Sobolev spaces

In this paper, we consider special classes of strongly convex functionals in Sobolev spaces. It is proved that functionals from such classes have the so-called H-property: weak convergence of sequences of arguments and convergence of such sequences with respect to a given functional imply strong convergence.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 378–394.Original Russian Text Copyright © 2005 by A. S. Leonov.This revised version was published online in April 2005 with a corrected issue number.     

On Morrey’s estimate of the Sobolev norms of solutions of elliptic equations

We give a complete proof of Morrey’s estimate for the W ^1,p -norm of a solution of a second-order elliptic equation on a domain in terms of the L ₁-norm of this solution. The dependence of the constant in this estimate on the coefficients of the equation is investigated.     

Embedding of Sobolev space in Orlicz space for a domain with irregular boundary

In this paper, we establish the embedding of a weighted Sobolev space in an Orlicz space for a domain with irregular boundary. We find an estimate of the order of growth of the N-function (defining the Orlicz space) and show that, under certain additional constraints on the weights, this estimate is sharp. We also establish the embedding in the space of continuous functions.     

具有奇异项的拟线性椭圆型方程组无穷多弱解的存在性

本文在一定条件下讨论了一类具有奇异项的,被两个pLaplacian算子控制的拟线性椭圆型方程组Dirichlet问题无穷多弱解的存在性.     

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.