Periodic and Bloch Solutions to a Magnetic Nonlinear Schrödinger Equation
Periodic and Bloch Solutions to a Magnetic Nonlinear Schrödinger Equation
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DOI：
10.1515/ans-2009-0404
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发表时间：
2009-11
影响因子：
1.8
通讯作者：
M. Clapp;R. Iturriaga;A. Szulkin
中科院分区：
文献类型：
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作者：
M. Clapp;R. Iturriaga;A. Szulkin
本文研究了方程（-i ∈ A）（-i ∈ A + A）2u + Vu =| u| p-2u，其中A ∈C1，α（<$N，<$N）和V ∈ C 0，α（<$N）对每个变量都是2π周期的，V > 0;且p ∈（2; 2*），其中2*：= ∞，如果N = 2，我们讨论两个问题：第一，2π周期解u：<$N → C的规范依赖性问题;第二，Bloch解的多重性问题.不像非周期情况下，问题（A）基本上与A无关（它是规范不变的），在周期情况下，这是远远不是真的。在对A的某些假设下，证明了：如果（A）的2π周期解与（A + z）的2π周期解之间存在一一对应关系，则z位于N的测度为零的子集中.利用这一事实证明了具有真实的拟动量的不可数Bloch解的存在性。
Abstract We study the equation (℘A) (-i▽ + A)2u + V u = |u|p-2u, where A ∈C1,α(ℝN,ℝN) and V ∈C0,α(ℝN) are 2π-periodic in each variable, V > 0; and p ∈(2; 2*) with 2* := ∞ if N = 2 and We address two questions: first, the gauge-dependence problem for 2π-periodic solutions u : ℝN → C and second, the multiplicity of Bloch solutions. Unlike the nonperiodic case where problem (℘A) is basically independent of A (it is gauge invariant), in the periodic case this is far from being true. Under some assumptions on A we show that, if there exists a one-to-one correspondence between the 2π-periodic solutions of (℘A) and those of (℘A + z) preserving their absolute value, then z lies in a subset of measure zero of ℝN. We use this fact to show the existence of an uncountable set of Bloch solutions with real quasimomentum.