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.     

一类具临界指数和等值面边值条件椭圆型方程解的存在性

本文利用临界点理论给出了RN(N≥3)中有界光滑区域上的拟线性椭圆型方程-△pU=|u|p*-2u a(x)|u|p-2u f(x,u),X∈Ω(P*=Np/(N-p),1相似文献   

Neumann Problem of Quasilinear Elliptic Equations with Limit Nonlinearity in Boundary Condition

This paper is concerned with the existence of solutions to the equation D_j(a^{ij}(x,u)D_i,u)-\frac{1}{2}D_sa^{ij}(x,u)D_iuD_ju + λ u = 0 on a bounded domain under the Neumann boundary condition a{ij}(x,u)D_iuϒ_j = lul^{\frac{2}{n-2}}u.     

在环域内一类拟线性椭圆型方程正对称解的存在性

利用Ｌｅｒａｙ－Ｓｃｈａｕｄｅｒ不动点定理和变分法得到了边值问题正对称解的存在性,这里　是ＩＲ~Ｎ中的环城．     

Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the nonlinearities.     

具Neumann边界条件的拟线性椭圆方程组的多解存在性

通过运用Ricceri的一个三临界点定理，得到了一类具变分结构的拟线性椭圆方程组的多解的存在性．     

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。     

Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

Regularity results and critical group estimates are studied for critical (p, r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.     

R~N空间中退化的logistic型拟线性椭圆方程正解存在唯一性

本文研究下列退化的logistic型p-Laplacian方程:-△Apu=a(x)|u|p-2u- b(x)|u|q-1u,x∈RN(N≥2)．在对系数a(x),b(x)在无穷远处的性质加以一般限制,得出了正解唯一存在性定理．我们的结果改进了文[1]和[2]中的相应结果．     

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.     

具有Dirichlet边界条件的一类拟线性椭圆方程组的多重解的存在性

运用Ricceri三临界点定理,研究了一类具有Dirichlet边界条件的拟线性椭圆方程组问题,证明了该方程组在其非线性项满足某些新的条件时至少存在三个解.     

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.     

On the Existence of Positive Solutions of Nonlinear Elliptic Equations

We study the existence of positive solutions of the nonlinear elliptic problem in D with u=0 on D, where and are two Randon's measures belonging to a Kato subclass and D is an unbounded smouth domain in ^d(d3). When g is superlinear at 0 and 0f(t)t for t(0,b), then probabilistic methods and fixed point argument are used to prove the existence of infinitely many bounded continuous solutions of this problem.     

带Robin边界条件的拟线性抛物方程解的整体存在性与爆破

主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界.     

拟线性椭圆型方程带有混合奇异项的正整体解

在本文中,研究了方程div(｜u｜p-2u) f(x,u)=0,x∈RN,N≥3的正整体解,其中f(x,u)在u=0未假定是正则的,且f(x,u)可以同时包含超线性,亚线性项和奇异项.     

On the Existence of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

We study the existence of positive radial solutions for a class of quasilinear elliptic systems in a ball domains via the blowing up argument and degree theory. The main results of the present paper are new and extend the previously known results.     

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.     

Existence and Uniqueness of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

This article is concerned with the existence and uniqueness of positive radial solutions for a class of quasilinear elliptic system. With some reasonable assumptions on the nonlinear source functions and their coefficients, the existence and the upper and lower bounds of the positive solutions will be provided by using the fixed point theorem and the maximum principle for the quasilinear elliptic system.     

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.     

拟线性方程解的存在性

利用一个推广的Landesman-Lazer型条件获得了一类拟线性椭圆方程的解的存在性结果．