R3 雙臨界Kirchhoff型方程正解的存在性

宋雅倩， 張福偉， 劉進(jìn)生

(太原理工大學(xué) 數(shù)學(xué)學(xué)院， 山西 太原 030024)

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R3雙臨界Kirchhoff型方程正解的存在性

宋雅倩， 張福偉， 劉進(jìn)生

(太原理工大學(xué) 數(shù)學(xué)學(xué)院， 山西 太原 030024)

利用變分方法研究了R3上具有雙臨界非線性項(xiàng)的Kirchhoff型方程正解的存在性. 首先證明了該問題的能量泛函滿足山路引理的幾何條件，從而證明了能量泛函存在(PS)c序列，進(jìn)而通過(PS)c序列的有界性與弱極限的非平凡性及徑向?qū)ΨQ空間的性質(zhì)證明了此(PS)c序列具有強(qiáng)收斂子列，因此證明了能量泛函存在非平凡臨界點(diǎn)，于是此問題存在非平凡解，最后證明了此非平凡解是正解.

Kirchhoff型方程； 雙臨界非線性項(xiàng)； 山路引理； 正解

0 引 言

本文主要考慮以下雙臨界Kirchhoff型方程

解的存在性. 其中常數(shù)a,b>0,2*= 6, 2*(s)=6-2s,s∈(0,1). 而

D1,2(R3)={u∈L2*(R3)|u∈L2(R3)},

其內(nèi)積與范數(shù)分別為

〈u,v〉=∫R3u·v,

由于Kirchhoff型方程的重要性，近年來，很多學(xué)者研究了如下的Kirchhoff型問題[1-8]

非平凡解的存在性. 同時(shí)，也有學(xué)者研究帶有Sobolev臨界指數(shù)的Kirchhoff 型方程[9-13]

文獻(xiàn)[14]運(yùn)用山路引理證明了RN中雙臨界p-Laplace方程

定理 1 對任意a,b>0,s∈(0,1), 方程(1)在D1,2(R3)中至少存在一個(gè)正解.

1 主要結(jié)果的證明

在徑向空間

R3)={u∈D1,2(R3)|u(x)=u(|x|)}

中考慮問題(1). 由文獻(xiàn)[14]知

引理 1 能量泛函φ滿足山路引理的幾何條件. 即

1)φ(0)=0， 并且存在α,ρ>0， 當(dāng)‖u‖=ρ時(shí)， 有φ(u)≥α>0.

注意到2*(s)=6-2s,s∈(0,1)， 從而存在α,ρ>0， 當(dāng)‖u‖=ρ時(shí)， 有φ(u)≥α>0.

從而存在充分大的t0>0， 使得‖t0u‖>ρ且φ(t0u)<0. 令e=t0u， 則‖e‖>ρ且φ(e)<0.

φ(γ(1))<0},

即{un}是泛函φ的一個(gè)(PS)c序列.

為了證明φ滿足(PS)c條件， 本文在引理2～引理8中研究了(PS)c序列的結(jié)構(gòu).

引理 2 若{un}為φ的(PS)c序列， 則{un}有界.

證明 由于φ(un)→c,φ′(un)→0且4<2*(s)<6， 則

故{un}有界.

對任意的0

A={x∈R3|r≤|x|

C2∫A|vn-v|2→0,

∫R3|un||(η2)||un|≤

式中：C(η)為正常數(shù)[14].

∫R3|(η un)|2=∫R3|ηun|2+o(1).

由引理2， {un}有界. 故

〈φ′(un,η2un〉=o(‖η2un‖)=o(‖un‖)=

o(1), n→∞.

由式(5)～式(8)得

o(1)=〈φ′(un),η2un〉=a∫R3|(η un)|2+

b∫R3|un|2∫R3|(η un)|2+o(1),

從而

‖η un‖2=o(1),

對任意的δ>0， 令

由引理4可知， α,β,γ的取值與δ無關(guān).

o(1)=

a∫Bδ(0)|un|2+b∫R3|un|2∫Bδ(0)|un|2-

從而

a∫Bδ(0)|

由α,β,γ定義可知αγ≤α+β. 于是結(jié)論成立.

所以

這與引理7矛盾.

,φ〉=∫R3aφ+

bB2∫R3φφ.

所以

bB2∫R3|

矛盾. 故∫R3|||2， 即又由于

b∫R3||2∫R3

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Existence of Positive Solutions for Kirchhoff Type Problems in R3with Multiple Critical Nonlinearities

SONG Ya-qian, ZHANG Fu-wei, LIU Jin-sheng

(College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China)

The existence of positive solutions for Kirchhoff type problem with multiple critical nonlinearities was investigated by using variational method. Firstly, it was proved that the energy functional possessed mountain pass geometry and got a (PS)csequence.Then, it was demonstrated that the (PS)csequence contained strong convergent subsequence through the boundedness of the (PS)csequence, the non triviality of weak limit, and the property of radial symmetry space. Therefore, the energy functional has at least one nontrivial critical point and the problem has at least one nontrivial solution. Consequently, it is achieved that the nontrivial solution is positive.

Kirchhoff type problem; multiple critical nonlinearities; mountain pass theorem; positive solution

1673-3193(2016)06-0576-05

2016-05-28

宋雅倩(1990-)， 女， 碩士生， 主要從事非線性泛函分析研究.

張福偉(1957-)， 女， 副教授， 主要從事非線性泛函分析研究.

O175.2

A

10.3969/j.issn.1673-3193.2016.06.005