Existence and Multiplicity of Radial Positive Solutions for a Quasilinear Elliptic Equation with Sign-Changing Nonlinearity

In this paper we will be concerned with questions of existence and multiplicity of radial nonnegative solutions of the quasilinear elliptic equation We will use variational methods in order to prove the existence of multiple solutions in case f is a sign-changing nonlinearity.     

A Priori Estimates and Existence of Positive Solutions for a Quasilinear Elliptic Equation

On the basis of some new Liouville theorems, under suitableconditions, a priori estimates are obtained of positive solutionsof the problem where R^N (N 2) is a bounded smooth domain, p>1 and isa parameter, , q are given constants such that p–1<<p*–1, <q, p*=Np/(N–p) if N > p and p*= when N p, and a(x) is a continuous nonnegative function. Makinguse of the Leray–Schauder degree of a compact mappingand a priori estimates, the paper finds that the problem abovepossesses at least one positive solution. It also discussesthe corresponding perturbed problem, where a(x) is replacedby a(x)+, >0. The results are strikingly different from thoseobtained for the case =p–1.     

一类拟线性椭圆方程的多重解

利用变分法和一个三临界点定理,证明了一类拟线性椭圆方程-div(a|u|p)|u|p-2u)=λf(x,u),u=0,ΩΩ,在某些新的条件下至少存在三个解,其中ΩRn(n≥1)是一个具有光滑边界的有界区域,且a∈C(R ,R),p>n,λ>0为一实参数.并给出了该结论在毛细现象中的广义Capillarity方程的一个应用.     

一类奇异拟线性椭圆方程正解的多重性

研究奇异拟线性椭圆型方程{-div(|x|~(-ap)|▽u|~(p-2)▽u) + f(x)|u|~(p-2) = g(x)\u|~(q-2)u + λh(x)|u|~(r-2),x R~N,u(x) 0,x∈ R~N,其中λ0是参数,1pN(N3),1rpgp*=0a(N—p)/p,p*=Np/{N~pd),aa+l,d=a+l-60,权函数f(x),g(x),h(x)满足一定的条件.利用山路引理和Ekeland变分原理证明了问题至少有两个非平凡的弱解.     

具有变号非线性项的拟线性椭圆方程多解的存在性

利用山路定理研究了具有Dirichlet边值问题-div(|▽u|~(2p-2)▽u/(1+|▽u|~(2p))~(1/2))=λf(x,u)多解的存在性,且非线性项f可以变号.     

拟线性椭圆型议程组正整体解的存在性

本文讨论了一类满足初值条件的非线性椭圆型方程组正整体解的存在性.     

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．     

Infinite Multiplicity of Positive Entire Solutions for a Semilinear Elliptic Equation

We establish that the elliptic equation Δu+K(x)u^p+μf(x)=0 in Rⁿ has infinitely many positive entire solutions for small μ?0 under suitable conditions on K, p, and f.     

一类拟线性椭圆型方程组的正对称解的存在性

本文讨论了球域上一类拟线性椭圆型方程组正对称解的存在性和多解性     

一类拟线性椭圆型方程正对称解的先验界估计

本文讨论了问题－ ｄｉｖ（｜Ｄｕ｜ｐ－ ２Ｄｕ） ＝ λｆ（ｕ）,　　在Ω中,ｕΩ＝ ０正对称解的先验界估计     

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。     

椭圆边值系统的正径向解的存在性与多解性

通过利用锥拉伸及锥压缩型的Krasnosel‘skii不动点定理，我们研究了一类椭圆边值系统的正径向解的存在性，非存在性与多解性。     

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.     

Existence of Solutions to a Singular Elliptic Equation

We study the equation ${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}}${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}} in Ω with Dirichlet boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term 1/u ^β near u ~ 0 by using a function g _ε (u) which pointwisely tends to 1/u ^β as ε → 0. When the parameter λ > 0 is large enough, the corresponding energy functional has critical points u _ε . Letting ε → 0, then u _ε converges to a solution of the original problem, which is nontrivial, nonnegative and vanishes at some portion of Ω. There are two nontrivial solutions.     

Structure of Positive Solutions of Quasilinear Elliptic Equations with Critical and Supercritical Growth

The structure of positive radial solutions to a class of quasilinear elliptic equations with critical and supercritical growth is precisely studied. A large solution and a small solution are obtained for the equations. It is shown that the large solution is unique, its asymptotic behaviour and flat core are also discussed.     

Existence and Multiplicity Results for a Degenerate Elliptic Equation

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.     

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.     

A Priori Estimates and Existence of Positive Solutions to Quasilinear Elliptic Equations in General Form

In this paper we prove the existence of a positive solution to the following superlinear elliptic Dirichlet problem, - Σ^n_{i,j=1}aij(x, u, Du)D_{ij}u = f(x, u, Du) in Ω, \quad u = 0 on ∂Ω where f satisfies certain growth conditions.