Asymptotic analysis and inverse problems of the nonlinear elliptic eigenvalue problems

本文研究了非线性椭圆方程的特征函数和特征值的精确渐近性质，并研究了与其相关的特征值反问题。阐明了一类非线性项方程分岔曲线的整体结构和局部结构。对于逆问题，我们主要研究了logistic型方程的逆分岔问题。利用常微分方程理论和分岔曲线的渐近展开公式，利用分岔曲线的渐近性质确定了未知的非线性项。

In this study, we investigated the precise asymptotic properties of the eigenfunctions and eigenvalues of nonlinear elliptic equations, and studied the inverse eigenvalue problems associated with thedirect problems. We clarify the global and local structures of the bifurcation curves for the equations with several types of nonlinear terms. As for inverse problems, we mainly studied the inverse bifurcation problems for logistic type equations. By using the theory of ordinary differential equations and the asymptotic expansion formulas for bifurcation curves, we determined the unknown nonlinear terms by the asymptotic behaviors of the bifurcation curves.