各向异性泛函与各向异性方程解的局部有界性

证明了各向异性泛函的极小点与各向异性方程     

一类似线性椭圆型方程在各向异性Соболев空间中非平凡解的存在性

本文借助没有Ｐ．Ｓ．条件的广义越山引理证明了一类拟线性椭圆型方程在各向异性Соболев空间中的非平凡解的存在性。     

具有临界Sobolev指数的拟线性椭圆型方程的一个全局紧性结果

本文得到了下面泛函的（ＰＳ）。序列的一个全局紧性结果：其中Ω是ＲＮ中的一个有界区域，２≤ｐ＜Ｎ，ｐ＊＝ｐＮ／（Ｎ－ｐ）．     

一类拟线性椭圆型方程在各向异性Соболев空间中非平凡解的存在性

本文借助没有P.S.条件的广义越山引理证明了一类拟线性椭圆型方程在各向异性空间中的非平凡解的存在性。     

临界增长的椭圆型方程广义解的有界性及先验估计

本文证明了椭圆型方程由divA（x，u,u）=B(x,u,u)(在G内)在带权soblev空间中的广义解的有界性，并做出了有界解最大模的先验估计．A（x，u，ζ)和B（x，u，ζ)关于u的增长阶达到了临界指数，而且允许B（x,u,ζ)关于ζ的增长阶γ满足自然增长条件，从而推广和改进了已有的所有结果．     

拟线性椭圆型方程的非线性边值问题——Sobolev临界指数的情形

本文讨论一类具有Ｓｏｂｏｌｅｖ临界指数的拟线性椭圆型方程非线性边值问题的非平凡解的存在性,利用集中紧原理得到一个存在性结果。     

各向异性增长拟线性椭圆型方程广义解最大模的先验估计

本文我们第一次给出了几类各向异性增长拟线性椭圆型方程广义解的最大模的先验估计。     

半线性次椭圆型方程和满足Hormander条件的向量场的Sobolev不等式

本文研究以Ｈｏｒｍａｎｄｅｒ平方和算子为主部的半线性次椭圆型方程的Ｄｉｒｉｃｈｌｅｔ问题的解的存在性和正则性，问题来源于所谓次椭圆几何学理论．与此同时，对于满足Ｈｏｒｍａｎｄｅｒ条件的一组向量场，我们证明了相应的Ｓｏｂｏｌｅｖ不等式．这一结果进一步的完善了Ｈｏｒｍａｎｄｅｒ平方和算子与Ｌａｐｌａｃｅ算子的类比研究工作．     

无界域上具有Sobolev临界指数的椭圆型方程解的存在性

本文利用山路引理和集中紧原理研究无界域上具有Sobolev临界指数的一类拟线性椭圆型方程的非平凡解的存在性。     

组合KdV方程解在图灵可计算意义下的有界性

运用数学归纳法,Gronwall不等式及方程的守恒量等工具,研究组合KdV方程初值问题解的有界性.首先在schwartz空间得到了方程解及解的任意阶导的上确界可以由初值为变量的图灵可计算函数来控制,由于schwartz空间S(R)是Sobolev空间Hs(R)(s≥0)的稠子空间,结果可以直接推广到sobolev空间Hs(R)(s≥0),所以组合KdV方程解在Hs(R)(s≥0)上确界可以由一个可计算函数来控制,从而为研究解算子的可计算性并运用图灵机计算组合KdV方程的解奠定了基础.     

Positive Solutions of Quasilinear Elliptic Equations

This paper is concerned with existence theorems for positive solutions of the Dirichlet problem for quasilinear elliptic differential equation containing a gradient term. Using the shooting method and the a priori estimates for the first zero, we obtain sufficient conditions for the existence of classical positive solutions of the problem in the ball.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 202–211.Original Russian Text Copyright © 2005 by E. I. Galakhov.     

Generating Singularities of Solutions of Quasilinear Elliptic Equations Using Wolff's Potential

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of p-Laplacian type. If p < < N and the right-hand side is a Radon measure with singularity of order at x_0A , then any supersolution in W^1,p() has singularity of order at least (–p)/(p–1) at x₀. In the proof we exploit a pointwise estimate of A-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff's potential of Radon's measure.     

Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents

Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.     

Existence of Solutions for Some Quasilinear Degenerate Elliptic Inclusions in Weighted Sobolev Spaces

We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form ?div(𝒜(x, u, ?u)) + f(x)g(u) ∈ H(x, u, ?u), where 𝒜(x, u, ?u) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H(x, u, ?u) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained.     

一类带临界Sobolev指数及有拟超临界Neumann边界条件的椭圆方程正解的多重性

本文讨论了Ω上如下一类带临界增长的椭圆方程在拟超临界的Neumann边界条件下正解的存在性:-Div(| u |p-2 u) =λum up*-1,-| u |p-2 u ν=ψ(x)uq-1,x∈Ω,x∈Ω.这里Ω∈RN,(N≥3)是光滑有界区域, 1≤p < N,0< m < p-1,(N -1)pN - p= p*N-1 ≤q < p*,其中p* =NpN - p是W1,p(Ω)→Ls(Ω)的Sobolev临界指数,p*N-1 =(N -1)pN - p是W1,p(Ω)→Lt( Ω)的在(N-1)维流形上的临界指数,λ>0是一个正参数.     

The Boundedness for Generalized Solutions of Quasilinear Elliptic Equations

Under the appropriate conditions on u, the generalized solution of the elliptic equation ∫_G {∇v ⋅ A(x, u, ∇u) + vB(x, u, ∇u)}dx = 0, \quad ∀v ∈ {\WW}¹_p(G) ∩ L_∞(G) for which even the natural growth condition p(1 - 1/p^∗) < ϒ < p is permitted, the local and global boundedness of u are proved.     

含Hardy位势的拟线性椭圆型方程非平凡解的存在性

设0∈Ω∈R^N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|²)=μu/|x|2)=μu/|x|²+λg(x,u),x∈Ω,其中μ>0,λ>0为常数,g(x,u)为Caratheodory函数.     

泛函极小与非线性椭圆方程解的局部正则性

利用Giaquinta和Giusti的嵌入不等式和Sobolev空间方法,证明了在更一般条件下的变分问题中的泛函极小与非线性椭圆方程弱解的局部正则性.     

一类拟线性椭圆型方程下解的先验估计

本文在一定的结构条件下,对如下形状的拟线性椭圆型方程∫Ｇｖ·Ａ（ｘ,ｕ,ｕ）＋ｖＢ（ｘ,ｕ,ｕ）｛｝ｄｘ＝０,ｖ∈Ｗ１ｐ（Ｇ）的下解作出了新的先验估计,发展了Ｇｉｌｂａｒｇ＿Ｔｒｕｄｉｎｇｅｒ的某些结果·     

环域上一类拟线性椭圆方程正的径向解的多重性

研究了方程-div(‖Du‖^-2Du)=λf(u)在R^n,n≥2中环域上的正的径向解的多重性。当f在正区域上有多个峰的情况下,我们获得了多个解。