周 斌，何 丹

(湖南工學(xué)院 基礎(chǔ)課部, 湖南 衡陽(yáng), 421002)

帶雙井勢(shì)函數(shù)的一維p-Laplace方程解對(duì)應(yīng)層的位置

周 斌，何 丹

(湖南工學(xué)院 基礎(chǔ)課部, 湖南 衡陽(yáng), 421002)

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

p-Laplace；雙井勢(shì)函數(shù)；n-模解；層

Allen-Cahn 方程是一個(gè)著名的兩相過(guò)渡模型,對(duì)于一維的Allen-Cahn 方程多層解的相關(guān)性態(tài), Kimie Nakashima在文獻(xiàn)[1-2]中已經(jīng)給出了詳盡的討論.

本文是Allen-Cahn方程的一個(gè)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(wèn)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(shè)(y)在y=0附近有一個(gè)零點(diǎn)，將y=0附近最右端的零點(diǎn)記作yk，將a?看作0，重復(fù)命題2的證明過(guò)程，即得證. 當(dāng)α( y)≤0時(shí)，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(shì)函數(shù)的一維p-Laplace方程解的零點(diǎn)分布[J]. 邵陽(yáng)學(xué)院學(xué)報(bào)： 自然科學(xué)版, 2010, 7(3)：6-8.

[4] 周斌, 何丹. 兩相模型的導(dǎo)數(shù)估計(jì)[J]. 經(jīng)濟(jì)數(shù)學(xué), 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?yàn)閼?yīng)用數(shù)學(xué).