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.     

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．     

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 Properties of Radial Solutions of a Sub-linear Elliptic Equation

A non lipschitzian nonlinear elliptic equation is reviewed and results of existence, uniqueness, positivity and classification are proved using direct methods derived from the equation.     

On the Existence of Positive Solutions for a Class of Semilinear Elliptic Systems

We study the existence of positive solutions for a class of semilinear ellipcic systems in general domains via the blow up argument and degree theory. The main idea can be used to establish the existence of positive solutions for the Navier problems of polyharmonic semilinear equations in general domains.     

On a Class of Quasilinear Elliptic Equations

We consider a class of quasilinear elliptic boundary problems, including the following Modified Nonlinear Schrödinger Equation as a special case: $$\begin{cases} ∆u+ \frac{1}{2} u∆(u^2)−V(x)u+|u|^{q−2}u=0 \ \ \ in \ Ω, \\u=0 \ \ \ \ \ \ \ ~ ~ ~ on \ ∂Ω, \end{cases}$$ where $Ω$ is the entire space $\mathbb{R}^N$ or $Ω ⊂ \mathbb{R}^N$ is a bounded domain with smooth boundary, $q∈(2,22^∗]$ with $2^∗=2N/(N−2)$ being the critical Sobolev exponent and $22^∗= 4N/(N−2).$ We review the general methods developed in the last twenty years or so for the studies of existence, multiplicity, nodal property of the solutions within this range of nonlinearity up to the new critical exponent $4N/(N−2),$ which is a unique feature for this class of problems. We also discuss some related and more general problems.     

Blow Up in a Semilinear Wave Equation

We consider a semilinear wave equation of the form u_tt(x, t) - Δu(x, t) = - m(x, t)u_t(x, t) + ∇Φ(x) ⋅ ∇u(x, t ) + b(x)|u(x, t)|^{p-2}u(x, t) where p > 2. We show, under suitable conditions on m, Φ, b, that weak solutions break down in finite time if the initial energy is negative. This result improves an earlier one by the author [1].     

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.     

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.     

一类非线性耗色散方程解的Blow—up

本文研究一类四阶非线性耗、色散波动方程的补边值问题,在一定条件下,得到了方程解的blow up性质。     

On a Quasilinear Elliptic Equation with Superlinear Nonlinearities

This work is devoted to studying a quasilinear elliptic boundary value problem with superlinear nonlinearities in a weighted Sobolev space in a domain of $\mathbb{R}^{N}$. Based on the Galerkin method, Brouwer's theorem and the weighted compact Sobolev-type embedding theorem, a new result about the existence of solutions is revealed to the problem.     

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.     

具有强阻尼项和非线性阻尼项的波动方程解的整体存在性和有限时间爆破

本文考虑了一类具有强阻尼和非线性阻尼项的波动方程:utt, - △u - ω△u1, +μ | u1|m-2 u1 =| u |p-2 u,其中P>2,m>2,w=μ1.利用变分法和紧性引理,本文证明了基态驻波解的存在性.并且得到了解的整体存在和爆破条件.     

一类拟线性椭圆型方程正奇异解的存在性

该文得到了一类拟线性椭圆型方程在球域( Ω=B_R={x∈R^N:|x|相似文献   

守恒相场模型弱解的Blow—up

本文讨论一类守恒相场模型弱解的性态,证明当ａ２ｐ－１＜０及初始数据充分大时解在有限时刻Ｂｌｏｗ－ｕｐ     

Ground State Solutions for a Semilinear Elliptic Equation Involving Concave-convex Nonlinearities

This work is devoted to the existence and multiplicity properties of the ground state solutions of the semilinear boundary value problem $-Δu=λa(x)u|u|^{q-2}+ b(x)u|u|^{2^∗-2}$ in a bounded domain coupled with Dirichlet boundary condition. Here $2^∗$ is the critical Sobolev exponent, and the term ground state refers to minimizers of the corresponding energy within the set of nontrivial positive solutions. Using the Nehari manifold method we prove that one can find an interval L such that there exist at least two positive solutions of the problem for $λ∈Λ$.     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

Large and Small Solutions of a Class of Quasilinear Elliptic Eigenvalue Problems

Existence and uniqueness results for large positive solutions are obtained for a class of quasilinear elliptic eigenvalue problems in general bounded smooth domains via a generalization of a sweeping principle of Serrin. The nonlinear terms of the problems can be negative in some intervals. The existence and structure of a mountain pass solution are also discussed. We show that this solution develops to a spike layer solution.     

Blow Up for Solutions of Nonlinear Doubly Degenerate Parabolic Equation

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