李 瑞

(山西大學(xué) 數(shù)學(xué)科學(xué)學(xué)院 山西 太原 030006)

一類(lèi)非局部(p,q)-Laplace方程非負(fù)解的存在性

李 瑞

(山西大學(xué) 數(shù)學(xué)科學(xué)學(xué)院 山西 太原 030006)

研究了RN中一類(lèi)帶有非局部項(xiàng)的(p,q)-Laplace方程非負(fù)解的存在性.在f(x,t)滿足一定條件下,得到能量泛函Cerami序列的有界性,結(jié)合變分法證明了非負(fù)解的存在性.

(p,q)-Laplace方程； 非局部項(xiàng)； Cerami條件； 非負(fù)解

0 引言

考慮帶有非局部項(xiàng)的(p,q)-Laplace方程，

(1)

(2)

當(dāng)p=q>1時(shí),方程(2)為單個(gè)p-Laplace方程.當(dāng)p=q=2時(shí),方程(2)為非線性Laplace方程

-Δu+au=f(x,u),x∈RN.

(3)

方程(2)是一般的反應(yīng)擴(kuò)散方程的穩(wěn)定態(tài)情形，ut=div[D(u)u]+f(x,u),其中：u代表濃度；D(u)q-2，是擴(kuò)散系數(shù)；f(x,u)是反應(yīng)項(xiàng).這類(lèi)方程在物理、生物和化學(xué)中都有廣泛的應(yīng)用[1-4].文獻(xiàn)[5]研究了方程(3)正解的存在性,a>0為常數(shù).由于(p,q)-Laplace算子是非齊次的,處理一般橢圓方程的方法不再適用,文獻(xiàn)[4,6-7]中作者運(yùn)用不同的方法得到了方程(2)非平凡解的存在性.文獻(xiàn)[4]考慮了(RN)的情形.文獻(xiàn)[6]考慮了1

本文探究帶有非局部項(xiàng)的(p,q)-Laplace方程非平凡解的存在性.其中f滿足如下條件:

(f1)f(x,t)是一個(gè)Carathéodory函數(shù)滿足:f(x,0)=0,當(dāng)t>0時(shí)F(x,t)>0.其中，

定理1 令2≤q

1 準(zhǔn)備工作

引理1Wp,a,h是自反的Banach空間,當(dāng)s∈[p,p*]時(shí),Wp,a,h→Ls(RN)為連續(xù)嵌入，當(dāng)s∈[p,p*)時(shí)為緊嵌入.Wq,b,g的性質(zhì)類(lèi)似可得[6].

注1 通過(guò)直接計(jì)算,并運(yùn)用H?lder不等式,對(duì)任意u∈Wp,a,h,線性泛函φ∈Ep,a,hφφ)∈R是有界的,且有.定義賦范空間W=Wp,a,h∩Wq,b,g,及范數(shù).由W的定義及引理1直接可得.

推論1W是自反的Banach空間,當(dāng)s∈[q,p*]時(shí),W→Ls(RN)為連續(xù)嵌入,當(dāng)s∈[q,p*)時(shí)為緊嵌入.

序列{un}?W稱為是C1泛函I:W→R的Cerami序列，如果滿足:

(C)I的任意Cerami序列在W中都有收斂子列.

用比較弱的(C)條件代替(PS)條件,定量形變引理仍成立[10].在假設(shè)(C)條件成立的情況下,山路定理也成立[11].因此,如果C1泛函I:W→R滿足(C)條件及山路幾何條件,則I在W中有一個(gè)臨界點(diǎn).

2 主要結(jié)果

定義

引理2I+滿足(C)條件.

證明 令{un}?W為I+的Cerami序列,則

(4)

其中K為某個(gè)不依賴于n的正常數(shù),且

(5)

(6)

(7)

記

(8)

故有

(9)

由式(7)、(8)得

(10)

(11)

從而可得

(12)

(13)

(14)

從而

由上式可得

(15)

而由式(10)可知

(16)

∫f+(x,un)(un-u)→0.

(17)

對(duì)任意u∈W,由注1線性泛函φ∈W〈-Δpu,φ〉∈R,φ∈W〈-Δqu,φ〉∈R,φ∈W〈hΨpu,φ〉∈R,φ∈W〈gΨqu,φ〉∈R都有界.在W中un弱收斂于u,從而有.由算子-Δp,-Δq,hΨp,gΨq的單調(diào)性可得

hΨp(u)+gΨq(u),un-u〉,

〈-aΔpun-bΔqun+hΨp(un)+gΨq(un),un-u〉-〈-aΔpu-bΔqu+hΨp(u)+gΨq(u),un-u〉=

〈-aΔpun,un-u〉-〈-aΔpu,un-u〉+〈-bΔqun,un-u〉-〈-bΔqu,un-u〉+〈hΨp(un),un-u〉-

引理3 存在R,ρ>0使得

證明 (i) 令r∈(2p,p*).由條件(f2)、(f5),對(duì)任意ε>0,存在Cε>0使得

則

定理1的證明 由引理3,I+滿足山路幾何條件，結(jié)合引理2,存在0≠u(mài)∈W,對(duì)任意φ∈W滿足

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(責(zé)任編輯：方惠敏)

Existence of Nonnegative Solution to a Class of (p,q)-Laplacian Equation

LI Rui

(SchoolofMathematicalSciences,ShanxiUniversity,Taiyuan030006,China)

The existence of nonnegative solution to a class of (p,q)-Laplacian equation in RNwas studied. Under the assumptions off(x,t), the boundedness of the Cerami sequence of energy function was obtained. Combining variational method, the existence of the nonnegative solution was extended.

(p,q)-Laplacian equation; nonlocal term; Cerami condition; nonnegative solution

2015-11-06

國(guó)家自然科學(xué)基金資助項(xiàng)目(11071149,11301313,11101250)；山西省自然科學(xué)基金資助項(xiàng)目(2014021009-1, 2015021007).

李瑞(1990—), 女,山西呂梁人，碩士研究生,主要從事非線性泛函分析與非線性微分方程研究，E-mail:rli1990@sina.com.

李瑞.一類(lèi)非局部(p,q)-Laplace方程非負(fù)解的存在性[J].鄭州大學(xué)學(xué)報(bào)(理學(xué)版)，2016，48(2)：5-10.

O175.2；O177

A

1671-6841(2016)02-0005-06

10.13705/j.issn.1671-6841.2015231