Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations
Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations
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DOI：
10.1112/plms/pds072
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发表时间：
2011-11
影响因子：
1.8
通讯作者：
J. Bellazzini;L. Jeanjean;Tingjian Luo
中科院分区：
文献类型：
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作者：
J. Bellazzini;L. Jeanjean;Tingjian Luo
本文研究了当p∈(103,6)时，一类Schrödinger-Poisson-Slater方程在∈(*)iψ +Δψ−(| x |−1*| ψ |2)ψ+| ψ |p−2ψ=0时，具有规定L2范数的驻波的存在性和不稳定性。为了得到这样的解，我们研究了能量泛函F(u)=12‖∇u‖L2(R3)2+14∫R3∫R3 | u(x) |2| u(y) |2| x - y |dx dy - 1p∫R3 | u |pdx的临界点S(c)={u∈H1(R)3：‖u‖L2(R3)2=c,c>}。对于所考虑的值p∈(103,6)，泛函F在S(c)上从下无界，临界点的存在性由S(c)上的一个山口论证得到。我们证明了当c > 0足够小时存在临界点，当c > 0不小时期望得到不存在的结果。关于动力学，我们证明了对于具有‖u0‖22=c的关联柯西问题的初始条件u0∈H1(∈3)，山口能级γ(c)给出了全局存在的阈值。同时也证明了驻波在山口能级上的强不稳定性。最后，我们将Schrödinger-Poisson-Slater方程与经典非线性Schrödinger方程进行了比较。
In this paper, we study the existence and the instability of standing waves with prescribed L2‐norm for a class of Schrödinger–Poisson–Slater equations in ℝ3 (*) iψt+Δψ−(| x |−1*| ψ |2)ψ+| ψ |p−2ψ=0 when p∈(103,6) . To obtain such solutions, we look into critical points of the energy functional F(u)=12‖ ∇u ‖L2(R3)2+14∫R3 ∫R3 | u(x) |2| u(y) |2| x−y |dx dy−1p∫R3 | u |pdx, on the constraints given by S(c)={ u∈H1(R)3:‖ u ‖L2(R3)2=c,c>0 }. For the values p∈(103,6) considered, the functional F is unbounded from below on S(c) and the existence of critical points is obtained by a mountain‐pass argument developed on S(c). We show that critical points exist provided that c > 0 is sufficiently small and that when c > 0 is not small a nonexistence result is expected. Regarding the dynamics, we show for initial condition u0∈H1(ℝ3) of the associated Cauchy problem with ‖ u0 ‖22=c that the mountain‐pass energy level γ(c) gives a threshold for global existence. Also, the strong instability of standing waves at the mountain pass energy level is proved. Finally, we draw a comparison between the Schrödinger–Poisson–Slater equation and the classical nonlinear Schrödinger equation.