半線性橢圓方程之解的存在性與多重性

Abstract

在第一部份, 考慮在圓柱體上半線性橢圓方程之解的對稱破壞, 令圓柱體長度為參數, 證明會產生對稱破壞在一群特定臨界值. 在第二部份, 考率慮非線性項為類線性之半線性橢圓方程, 解的存在性, 唯一性, 以及這些解的行為.

In part 1 we study the problem of symmetry-breaking of positive symmetric solutions of a semi-linear elliptic equation on finite cylinders with mixed type boundary conditions in two dimentions. Taking the length as a bifurcation parameter, we prove that there are asymmetric bifurcations at certain critical numbers, and obtain some global results. In part 2 we consider some semi-linear elliptic equations with asymptotic linear non-linearity and show the existence, uniqueness, and asymptotic behavior of these solutions.