李紅英， 佘連兵，廖家鋒

(1.遵義師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,貴州遵義　563002；2.六盤水師范學(xué)院數(shù)學(xué)系,貴州六盤水　553004)

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一類奇異p-Laplace方程正解的存在性

李紅英1， 佘連兵2，廖家鋒1

(1.遵義師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,貴州遵義563002；2.六盤水師范學(xué)院數(shù)學(xué)系,貴州六盤水553004)

研究一類奇異p-Laplace方程，利用極小化方法獲得該問(wèn)題的一個(gè)正解， 從而充實(shí)了奇異橢圓問(wèn)題解的理論.

奇異p-Laplace方程；橢圓問(wèn)題；極小化方法；正解

1　引言及預(yù)備知識(shí)

考慮奇異p-Laplace方程

(1)

奇異橢圓方程起源于各種應(yīng)用學(xué)科，例如核物理、氣體動(dòng)力學(xué)、流體力學(xué)、邊層理論以及非線性光學(xué)等.由于應(yīng)用學(xué)科的需要，20世紀(jì)70年代后期，奇異橢圓方程開始引起人們的關(guān)注.當(dāng)p=2,1

2　主要結(jié)果

證明我們分三步來(lái)完成定理1的證明.

(2)

(3)

因此，依據(jù)范數(shù)的弱下半連續(xù)性以及(2)式和(3)式可得

根據(jù)下確界的定義可得m≤I(u*)，從而m=I(u*).

(4)

因此，u*滿足-Δpu*≥0.結(jié)合u*≥0且u*不恒等于零以及強(qiáng)極大值原理[12]可得在Ω中u*(x)>0.

下面證明u*是問(wèn)題(1)的解.根據(jù)(4)式，兩邊同時(shí)除以t可得

根據(jù)控制收斂定理可得

(6)

對(duì)一切的x∈Ω，我們定義

(7)

(8)

(9)

即u*是問(wèn)題(1)的一個(gè)正解.

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(責(zé)任編輯馬宇鴻)

Existence of positive solutions for a class of singularp-Laplacian equations

LI Hong-ying1，SHE Lian-bing2,LIAO Jia-feng1

(1.School of Mathematics and Computational Science,Zunyi Normal College,Zunyi 563002,Guizhou,China;2.Department of Mathematics,Liupanshui Normal College,Liupanshui 553004,Guizhou,China)

A class of singularp-Laplacian equations is considered.By applying a minimax method,the existence of positive solutions for this problem is obtained,which enrished the theory of solutions for singular elliptic problems.

singularp-Laplacian equation；elliptic problem；minimax method；positive solution

10.16783/j.cnki.nwnuz.2016.04.001

2015-08-04;修改稿收到日期:2015-11-30

貴州省自然科學(xué)基金資助項(xiàng)目(LH[2015]7049)

李紅英(1984—)，女，四川南充人,講師,碩士.主要研究方向?yàn)榉蔷€性分析.

E-mail:honghongying2005@163.com

O 177.1

A

1001-988Ⅹ(2016)04-0001-04