非线性项在零点和无穷远处非渐进增长的高维P-Laplacian半拟线性问题的全局分歧  

Bifurcation for the High-dimensional p-Laplacian Half-quasilinear Problems with Non-asymptotic Nonlinearity at 0 and∞

作　　者：沈文国 纳仁花 包理群[2] SHEN Wenguo;NA Renhua;BAO Liqun(Department of Basic Courses,Lanzhou Institute of Technology,Lanzhou 73QQ5Q,China;Department of Electronic and Information Engineering,Lanzhou Institute of Technology,Lanzhou 730050,China)

机构地区：[1]兰州工业学院基础学科部,兰州730050 [2]兰州工业学院电子信息工程系,兰州730050

出　　处：《应用数学学报》2022年第1期59-71,共13页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11561038);甘肃省自然科学基金(编号:20JR5RA377);兰州工业学院‘开物’科研创新团队支持计划资助(编号:2018KW-03)。

摘　　要：本文将研究一般区域上高维p-Laplacian方程保号解的存在性:{u(x)=0,x∈ЭΩ,-div(φp(■u))=a(x)φp(u^(+)+β(x)φp(u^(-))+ra(x)f(u)),x∈Ω,其中Ω是R^(N)中一个有界且在其边界上光滑的区域,N≥2,1<P<+∞,φp(s)=|s|^(p-2)s,a(x)∈C(Ω,(0,+∞)),u^(+)=max{u,0},u^(-)=-min{u,0},a{x},β(x)∈C(Ω);f∈C(R,R)对于s>0,sf(s)>0成立.当f0■(0,∞)或f∞∈(0,∞)(其中f0=_(|s|→0)limf(s)/φp(s),f∞=_(|s|→+∞)limf(s)/φp(s)),且r≠0属于一定区间时,可以获得上述高维p-Laplacian方程保号解的存在性.我们用全局分歧技巧和连通序列集取极限的方法获得主要结果.In this paper,we study the existence of one-sign solutions for the following problem:{u(x)=0,x∈ЭΩ,-div(φp(■u))=a(x)φp(u^(+)+β(x)φp(u^(-))+ra(x)f(u)),whereΩis a bounded domain in R^(N) with a smooth boundary ЭΩ and N≥2,1<p<+∞,φp(u^(+)+β(x)φp(u^(-))+ra(x)f(u)),x∈Ω is positive,s≠0.We give the intervals for the parameter r≠0 which ensure the existence of one-sign solutions for the above high-dimensional p-Laplacian half-quasilinear problems if f0■(0,∞) or f∞∈(0,∞),where f0=_(|s|→0)limf(s)/φp(s),f∞=_(|s|→+∞)limf(s)/φp(s).We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

关 键 词：全局分歧 高维半拟线性问题 P-LAPLACIAN方程 保号解 

分 类 号：O175.8[理学—数学]