带深井位势双调和方程的解  

Solutions of biharmonic equations with steep potential wells

作　　者：郭玉霞[1] 唐仲伟[2] 汪路顺 Yuxia Guo;Zhongwei Tang;Lushun Wang

机构地区：[1]清华大学数学科学系,北京100084 [2]北京师范大学数学科学学院,北京100875

出　　处：《中国科学：数学》2019年第1期21-38,共18页Scientia Sinica：Mathematica

摘　　要：本文研究下述双调和方程极小能量解的存在性:?~2u+[λV (x)-δ]u=|u|_(p-2)u, x∈R^N,(0.1)其中N≥5,λ> 0. p是次临界或临界的Sobolev指标,即2 <p≤2**,这里2**=2N/N-4为临界的Sobolev指标, V (x)是非负连续的深井位势,其零集V^(-1)(0):={x∈R^N:V (x)=0}的内部int V^(-1)(0)是R^N中非空的有界光滑区域.令μ0为定义在int V^(-1)(0)中齐次边界条件下?~2的第一特征值.对任意的0 <δ<μ0,本文证明:当λ> 0充分大时,(0.1)存在一个在V^(-1)(0)附近的极小能量解.In this paper,we are concerned with the existence of least energy solutions for the following biharmonic equations:△^2u + [λV(x)-δ]u = |u|^(p-2)u, x ∈ R^N,(0.1)where N≥5, 2<p≤2N/(N-4), λ > 0 is a parameter, V(x) is a nonnegative potential function with nonempty interior part of the zero set int V^(-1)(0), 0 < δ < μ0 and μ0 is the principal eigenvalue of △^2 in the zero set int V^(-1)(0) of V(x). We prove that the equation(0.1) admits a least energy solution which is trapped near the zero set V^(-1)(0)for λ > 0 large enough.

关 键 词：极小能量解 双调和方程 深井位势 

分 类 号：O175[理学—数学]