Qualitative Theory of Nonlinear 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；(2)由椭圆型方程在无界区域的定性问题导出的常微分方程的振动问题。我们的结果如下研究了p-拉普拉斯算子的特征值和特征函数的渐近性质。我们得到了L*-Poincare不等式的最佳常数，以及特征值和特征函数的极限在弱意义上满足的一个极限方程研究了（A, *）上的Sturm-Liouville方程。假设一个强非振荡条件，我们得到了一个正的主特征值序列和相应的主特征函数关于抛物方程的初值问题，我们得到了一个新的初值的充分条件，该条件决定了抛物方程的解是否爆破。进一步，我们研究了解在吹唇时刻的渐近行为考虑[a， *]上的二阶半线性微分方程。我们选取了两类非振荡解（即主解和非主解），并证明了当λ从0到无穷变化时，这些解的0的数目如何变化的精确信息研究了一类具有Dirichelt边界条件的退化拟线性椭圆方程的分岔问题或非线性特征值问题。利用我们的估计，我们可以将leray - shader度理论应用于我们的问题，并得到了非平凡弱解的分岔研究了由两个或多个不同频率输入信号驱动的Van der Pol型方程的拟周期解。从数值分析的角度得到了存在唯一性结果。

We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) bound-ary value problems of quasilinear elliptic equations in a bounded domain ; (ii) oscillatory problem of ordi-nary differential equations which derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.[1] The asymptotic behavior of eigenvalues and eigenfunctions of p-Laplace operator is investigated. We obtain the best constant of L*-Poincare inequality, and a limit equation which the limits of eigenvalues and eigenfuncitons satisfy in a weak sense.[2] A Sturm-Liouville equation on (a, *) is examined. Supposing a strongly nonoscillatory condition, we obtain a sequence of positive princial eigenvalues and the corresponding principal eigenfunctions.[3] Concering an initial value problem of parabolic equation, we obtain a new sufficient condition on the initial value which determines the solution to blow up. Furthermore, we investigate the asymtotic behavior of the solution at the blowing lip time.[4] We consider a second order half-linear differential equation on [a, *]. We take up the two classes of nonoscillatory solutions (i.e., princilal solutions and non principal solutions), and show that precise information can be drawn as to how the number of zeros of these solutions changes as lambda varies from zero to infinity.[5] A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichelt boundary condition is studied. By virture of our estimates, we can apply the Leray-Shauder degree theory to our problem and obtain the bifurcation of nontrivial weak solutions.[6] Quasiperiodic solutions to Van der Pol type equations driven by two or more distinct freequency input signals are considered. The existence and uniqueness results are obtained from the viewpoint of numerical analysys.