Analysis on the nonlinear elliptic eigenvalue problems

研究了多参数非线性椭圆型特征值问题。我们主要关注的是在参数非常大时，通过变分方法和奇异摄动方法澄清特征值参数和相关特征函数的渐近性质。研究了具有两个纯幂非线性项的双参数常微分方程。利用几种变分方法，建立了特征值的几个渐近公式。我们还研究了与单摆问题相关的双参数问题。在Dirichlet边界条件下，建立了参数非常大时边界层解的精确渐近公式。所得公式与有关单参数单摆问题的公式完全不同。对于单参数问题，我们研究了与单摆问题相关的问题。首先研究了常微分方程，在Dirichlet边界条件下，建立了参数较大时边界层的精确渐近公式。然后将此结果推广到光滑有界区域上的非线性椭圆型特征值问题。研究了具有纯幂非线性的非线性椭圆型特征值问题，建立了其特征值在L-2框架下的渐近公式。我们还处理了有界域上的摄动单摆问题。已知当参数较大时，对应的解在域内几乎是平坦的。我们成功地建立了解的内部行为的精确渐近公式，以精确地理解当参数非常大时，解在区域内的平坦程度。这里得到的公式用非线性项精确地表示出来。

We studied nonlinear elliptic eigenvalue problems with several parameters. Our main concern was to clarify the asymptotic properties of eigenvalue parameters and associated eigenfunctions by using variational methods and singular perturbation approaches when a parameter was very large.We studied two-parameter ordinary differential equations with two pure power nonlinear terms. We established several asymptotic formulas for eigenvalues by using several variational approaches. We also studied the two-parameter problems which were related to the simple pendulum problems. We established the precise asymptotic formulas for the solutions with boundary layers when a parameter was very large under the Dirichlet boundary conditions. The formulas obtained here were totally different from those for the associated one-parameter simple pendulum problems.For problems with one-parameter, we studied the problems which were related to the simple pendulum problems. We first studied ordinary differential equations and established precise asymptotic formulas for the boundary layers when a parameter was large under the Dirichlet boundary conditions. We next extended this result to the nonlinear elliptic eigenvalue problems in a smooth bounded domain. We studied nonlinear elliptic eigenvalue problems with pure power nonlinearity and established the asymptotic formula for the eigenvalues in L-2 framework. We also treated the perturbed simple pendulum problems in a bounded domain. It is known that if parameter is large, then an associated solution is nearly flat inside the domain. We have succeeded to establish the precise asymptotic formulas for the interior behavior of the solutions to understand precisely how flat the solutions were inside a domain when a parameter was very large. The formulas obtained here were exactly represented by using the nonlinear term.