Studies on the Structure of Solutions of Degenerate Quasilinear Elliptic Equations

本文考虑了拟线性退化椭圆型方程在原点和无穷远处渐近地等价于p-Laplacian算子。得到了关于这类方程解的一些结果，如下所列.首先，我们得到了由平凡解和无穷解分支出的正解分支的结构。也就是说，正解在p-Laplacian的第一特征值处分别从零解和无穷远解分支出来，所考虑的方程分别在零和无穷远渐近地等于p-Laplacian。而且，当方程的主部在零点和无穷远处的增长阶相同时，这些分支是相同的.其次，研究了p-Lapalace算子的本征值和本征函数在幂p趋于无穷大时的渐近性态。得到了Poincare不等式在p=* 时的最佳常数，并进一步给出了一个极限特征值问题，刻画了这些特征值和特征函数的极限.虽然极限特征值问题中的方程是一个完全非线性椭圆型方程，但特征值和特征函数的一对极限在粘性意义下解了这个极限特征值问题.最后，我们考虑了定义在整个空间而不是有界区域上的方程。在这种情况下，在对非线性算子的势部分作某些假设的情况下，与I.中所述的结果类似的结果也是有效的。我们的问题是以后要放松这些限制。

We have considered quasilinear degenerate elliptic equations which are equal to p-Laplacians asymptotically at tile origin and infinity. Some results concerning to the aspects of solutions for the equations of this type are obtained as are listed below.1. First we have obtained the structure of the branches of positive solutions bifurcated from the trivial solution and tile infinity solution. Namely, positive solutions bifurcate from the zero solution and the infinity solution at the first eigenvalues of the p-Laplacians which the considered equation are equal to aymptotically at zero and infinity respectively. Furthermore, in the case when the growth orders of the principal part of the equation at zero and infinity, these branches are identical.2. Next the asymptotic behavior of eigenvalues and eigenfunctions of p-Lapalace operators as the power p tends to infinity has been investigated. We have obtained the best constant of Poincare's inequality at p=*, and further, given a limit eigenvalue problem which characterize tile limits of these eigenvalues and eigenfunctions. Althogh the equation ill the limit eigenvalue problem is a fully nonlinear elliptic equation, a pair of the limits of eigenvalues and eigenfunctions solves this limit eigenvalue problem in the sense of viscosity.3. Finally we have considered the equation defined in the whole space instead of a bounded domain. In this case, tile similar results as are stated in I.are also valid under some assumptions on the potential part of the nonlinear operator. It is our problem to loose these restrictions hereafter.