摘要
本文考虑如下分数阶Kirchhoff方程:{M(∫∫R^3×R^3|u(x)-u(y)|~2|x-y|3+2sdxdy)(-?)su(x)+V (x)u=f (u), x∈R^3,u∈H^s(R^3),其中M(t)=ε^(2s)a+ε^(4s-3)bt是Kirchhoff函数,3/4<s<1,ε>0是小参数,位势V是正连续函数且有全局极小,非线性项f连续且在无穷远处次临界增长.利用Ljusternik-Schnirelmann畴数理论,本文得到了正解个数与位势V全局极小集拓扑之间的关系,证明了当ε→0^+时,这些正解在H^s(R^3)中收敛到极限方程的基态解,且这些解集中在位势V的全局极小附近.此外也得到了解的衰减估计.
In this paper, we consider the following fractional Kirchhoff equation,M(∫∫R^3×R^3|u(x)-u(y)|^2/|x-y|^(3+2s)dxdy)(-△)^su(x) + V(x)u = f(u), in R^3,u ∈ H^s(R^3),where M(t) = ε^(2s)a + ε^(4s-3)bt is a Kirchhoff function, 0 < s < 1 and ε > 0 is a small parameter, the potential V is a positive continuous function which has the global minimum, and f is supercubic but subcritical at infinity.Using the Ljusternik-Schnirelmann theory, we relate the number of positive solutions with the topology of the set Λ := {x ∈ R^3: V(x) = inf V }, and we show that these positive solutions converge in H^s(R^3) to a ground state solution of the limit equation, and concentrate around the set Λ in the sense that the values of V at the maximum points of these solutions convergent to the global minimum of V as ε → 0^+. Moreover, the decay estimate of solutions is also established.
作者
顾光泽
吴鲜
余渊洋
赵富坤
Guangze Gu;Xian Wu;Yuanyang Yu;Fukun Zhao
出处
《中国科学：数学》
CSCD
北大核心
2019年第1期39-72,共34页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:11661083和11771385)
云南省中青年学术与技术带头人后备人才计划(批准号:2015HB028)资助项目
