Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential
Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential
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DOI：
10.4171/jems/407
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发表时间：
2013-07
影响因子：
2.6
通讯作者：
Jaeyoung Byeon;Kazunaga Tanaka
中科院分区：
文献类型：
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作者：
Jaeyoung Byeon;Kazunaga Tanaka
考虑R上的奇摄动椭圆型方程ε21u−V(X)u+f(U)=0，u(X)>；0，其中对任意x→∞RN，V(X)>；0均为∈。奇摄动问题在R，−|x|→∞U(X)=0，c>；0上有相应的极限问题1U Lim|x+f(U)=0，U(X)>；0。Berestycki-Lions在文[3]中找到了非线性项f存在极限问题解的几乎充分必要条件。人们一直在努力构造奇异摄动问题的解，在f上可能的一般条件下，集中在位势V的结构稳定的临界点附近。本文证明了在f∈C_1上Berestycki-Lions的最优条件下，存在一个集中在V的拓扑稳定正临界点附近的解，并用极大极小方法刻画了它的临界值。
We consider a singularly perturbed elliptic equation ε21u− V (x)u+ f (u) = 0, u(x) > 0 on R , lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ∈ RN . The singularly perturbed problem has corresponding limiting problems 1U − cU + f (U) = 0, U(x) > 0 on R , lim |x|→∞ U(x) = 0, c > 0. Berestycki–Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f . In this paper, we prove that under the optimal conditions of Berestycki–Lions on f ∈ C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods.