Qualitative Theory of Quasilinear Degenerate Elliptic Differential Equations

我们研究了与非线性椭圆型方程定性理论有关的问题：(I)有界区域上拟线性椭圆型方程的边值问题；(Ii)半线性抛物型方程的爆破现象；(Iii)由无界区域上椭圆型方程定性问题导出的常微分方程解。我们的研究结果如下所示。研究了一类非线性特征值问题的正解。对于拟线性椭圆问题(^*)-div(φ(|∇u|)∇u)=λf(x，u)在Ω上，u=0在∂Ω上，我们假定φ和f上的渐近条件如下：φ(T)~t^&lt；p_0-2&gt；，f(x，t)~t^&lt；q_0-1&gt；as t→0和φ(T)~t^&lt；p_1-2&gt；，f(x，t)~t^&lt；q_1-1&gt；as t→∞。亚非线性(p_0&gt；q_0)和超非线性(p_1&lt；q_1)的综合作用意味着0&lt；λ&lt；Λ2的(^*)至少存在两个正解。对于具有凸非线性的半线性抛物型方程的初值问题，我们得到了判定解爆破的充分条件。根据相关静态问题的上解和下解，给出了判定准则。研究了[α，∞]上的一个Sturm-Liouville方程。(1)具有一般非线性的高阶常微分方程及Kiguraze类正解的确定。(2)一阶四维常微分方程组的振动性问题。(3)二阶半线性常微分方程组在[α，∞]上的特征值问题。

We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) boundary value problems of quasilinear elliptic equations in a bounded domain ; (ii) blowup phenomena of seminilear parabolic equations ; (iii) solutions of ordinary differential equations derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.1. Positive solutions of a class of nonlinear eigenvalue problems are investigated. For a quasilinear elliptic problem (^*)-div (φ(|∇u|) ∇u)=λf(x, u) in Ω, u=0 on ∂Ω, we assume asymptotic conditions on φ and f such as φ(t)〜t^<p_0-2>, f(x, t)〜t^<q_0-1> as t→0 andφ(t)〜t^<p_1-2>, f(x, t)〜t^<q_1-1> as t→∞. The combined effects of sub-nonlinearity (p_0>q_0) and super-nonlinearity (p_1<q_1) imply the existence of at least two positive solutions of (^*) for 0<λ<Λ.2. Concerning an initial value problem of semilinear parabolic equations with convex nonlinearities, we obtain sufficient conditions which determine the solution to blow up. The criteria are formulated in terms of super-solution and sub-solution of the related stational problem.3. A Sturm-Liouville equation on [α, ∞) is examined. (1) Higher order ordinary differential equations with general nonlinearities and determination of the Kiguraze class of positive sotuions. (2) Oscillation problem of first order 4-dimensional system of ordinary differential equations. (3) Eigenvalue problem of second order half-linear ordinary differential equations on [α, ∞).