帶雙井勢函數(shù)的一維p-Laplace方程解對應層的位置

周 斌，何 丹

(湖南工學院 基礎課部, 湖南 衡陽, 421002)

帶雙井勢函數(shù)的一維p-Laplace方程解對應層的位置

周 斌，何 丹

(湖南工學院 基礎課部, 湖南 衡陽, 421002)

研究了一類帶雙井勢函數(shù)的一維p-Laplace方程解，主要討論了方程解對應層的位置情況. 解的零點與交換層是一一對應的，得出方程解對應的層出現(xiàn)在()h x的局部極值點附近. 在()h x的局部極小值點附近只可能出現(xiàn)一個交換層，而多層出現(xiàn)在()h x的局部極大值點附近.

p-Laplace；雙井勢函數(shù)；n-模解；層

Allen-Cahn 方程是一個著名的兩相過渡模型,對于一維的Allen-Cahn 方程多層解的相關性態(tài), Kimie Nakashima在文獻[1-2]中已經給出了詳盡的討論.

本文是Allen-Cahn方程的一個推廣，將Allen-Cahn方程中的Laplace算子換成p-Laplace算子后,著重討論了下列問題解的交換層位置的分布情況,即：

1 定理與命題

2 引理及定理的證明

這與式(11)矛盾. 故命題2得證.

命題3的證明 考慮α( y)≥0的情形[5-8]，假設(y)在y=0附近有一個零點，將y=0附近最右端的零點記作yk，將a?看作0，重復命題2的證明過程，即得證. 當α( y)≤0時，同理可證.

[1] Nakashima K. Stable transition layers in a balanced bistable equation[J]. Differential and Integral Equations, 2000, 13： 1025-1038.

[2] Nakashima K. Multi-layered stationary solutions for a spatially inhomogeneors Allen-Cahn equation[J]. J-Differential Equations, 2003, 191： 234-276.

[3] 周斌, 何丹. 帶雙井勢函數(shù)的一維p-Laplace方程解的零點分布[J]. 邵陽學院學報： 自然科學版, 2010, 7(3)：6-8.

[4] 周斌, 何丹. 兩相模型的導數(shù)估計[J]. 經濟數(shù)學, 2010, 27(3)： 24-27.

[5] Wong Fu-Hsiang. Uniqueness of Positive Solutions for Sturm-Liouville Boundary Value Problems[J]. Proceedings of The American Mathematical Society, 1998, 126：365-374.

[6] Lucio Damascelli. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results[J]. Nonlinerar Analysis, 1998, 15： 493-516.

[7] Gui C, Schatzman M. Symmetric quadruple phase transition[J]. Indiana University Mathematical Journal, 2008, 57： 781-836.

[8] Rabinowitz P H. Some global resulets for nonlinear eigenvalue problems [J]. J Funct Anal, 1971, 7： 487-513.

The locatin of the layers of the solution to a class of one-dimensional p-Laplace equation with double well potential

ZHOU Bin, HE Dan
(Basic Department, Hunan Institute of Technology, Hengyang 421002, China)

The problem of the location of the layers of the solution to a class of one-dimensional p-Laplace equation with double well potential was put forward. There is an one-one correspondence between the zeros and the layers of the solutions. The layers only appear near the local extreme points of ()h x. At most a single layer can appear near each local minimum point of ()h x, the multi-layers can appear near the local maximum point of ()h x.

p-Laplace; double well potential equation; n-mode solution; layers; distribution of the zeros

O 175.29

：A

1672-6146(2010)04-0011-03

10.3969/j.issn.1672-6146.2010.04.004

2010-08-04

湖南省教育廳科研課題(10C0586)

周斌(1979-), 男, 碩士, 研究方向為應用數(shù)學.