带有限位势的非线性Schrdinger方程组的无穷多变号解  

Infinitely many sign-changing solutions for nonlinear Schrdinger systems with finite potential well

作　　者：刘嘉荃[1] 刘祥清[2] 王志强[3,4] 

机构地区：[1]北京大学数学科学学院,北京100871 [2]云南师范大学数学学院,昆明650500 [3]天津大学应用数学中心,天津300072 [4]Department of Mathematics and Statistics, Utah State University

出　　处：《中国科学：数学》2016年第5期587-604,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11171171;11271331;11361077和11271201);云南省中青年学术和技术带头人培养(批准号:2015HB028)资助项目

摘　　要：本文考虑非线性Schrdinger方程组-?u j+λj(x)u j=k i=1β_(ij) u_i^2 u_j,x∈R^N,u_j(x)→0,当|x|→∞时,j=1,...,k,其中N=2,3,β_(ij)是常数,满足β_(jj)>0(j=1,...,k),β_(ij)=β_(ji)0(1≤i<j≤k),λ_j(j=1,...,k)是位势函数.首先考虑带强制位势的方程组,利用流不变集方法证明带强制位势的方程组有无穷多变号解;然后在位势λ_j具有一定渐近性质(见正文(V_1)–(V_4))时,通过集中紧性分析,证明带强制位势扰动方程组的解趋于原来有限位势的方程组的解,从而证明原方程组有无穷多变号解.In this paper, we consider the following nonlinear Schr¨odinger systems in R^N：-？u_j ＋ λ_j（x）u_j =k i=1β_（ij） u_i^2 u_j, x ∈ R^N,u_j（x） → 0, as ｜x｜ → ∞, j = 1,..., k,where λ_j（x）, j = 1,..., k are potential functions with finite depth, β_（ij） ＇s are constants satisfying β_（ij） = β_（ji）, β_（jj） 0 for j = 1,..., k, β_（ij） 0 for i = j. We prove that the systems have an unbounded sequence of sign-changing solutions provided that the potentials λj satisfy the following conditions（V_1）–（V_4）：（V_1） λ_j ∈ C^1（R^N, R）, λ_j（x）≥m 0, x ∈ R^N;（V_2） lim｜x｜→∞ λ_j（x） = λ_j＊ ＋∞;（V _3）？λ_j？（x）e^（α｜x｜）→ ∞, as ｜x｜ → ∞, for α 0, where  =x｜x｜, x ∈ R^N／{0};（V_4） There exists  1 such that ｜Dλ_j（x）｜≤？λ_j？（x）, x ∈ R^N, ｜x｜ .We construct the solutions by approximations of solutions of systems with coercive potentials. We first consider the case of coercive potential, and by using the method of invariant sets of flow we obtain the existence of infinitely many sign-changing solutions of the systems. Then if λj satisfies some certain asymptotic properties（see the conditions（V_1）–（V_4））, by using the concentration compactness analysis, we will prove that the solutions of the perturbed systems tend to solutions of the systems with finite potential, then we get infinitely many sign-changing solutions of the original problem.

关 键 词：非线性Schrdinger方程组 有限位势 流不变集方法 变号解 

分 类 号：O175[理学—数学]