Qualitative Theory of Quasilinear Degenerate Elliptic Differential Equations

本文研究了非线性椭圆型微分方程定性理论的相关问题：(1)有界区域上拟线性椭圆型方程的边值问题；（ii）半线性抛物方程的爆破现象；(3)由无界区域内椭圆型方程定性问题导出的常微分方程的解。我们的研究结果如下：研究了一类非线性特征值问题的正解。拟线性椭圆问题(^ *)div(φ(|∇u |)∇u) =λf (x, u)在Ω,在∂u = 0Ω,我们假设渐近条件等φ和fφ(t) ~ t ^ < p_0-2 >, f (x, t) ~ t ^ < q_0-1 > t→0和φ(t) ~ t ^ < p_1-2 >, f (x, t) ~ t ^ < q_1-1 >当t→∞。子非线性（p_0>q_0）和超非线性（p_1<q_1）的联合效应表明，对于0<λ<Λ.2,（^*）至少存在两个正解。关于一类具有凸非线性的半线性抛物型方程初值问题，得到了判定解爆破的充分条件。这些判据是用相关稳态问题的超解和子解来表示的。研究了[α，∞)上的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系统的振动问题。(3) [α,∞)上二阶半线性常微分方程的特征值问题。

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, ∞).