论文标题
关键的Schrödinger-Poisson系统的半经典解决方案，涉及多个竞争潜力
Semiclassical solutions for critical Schrödinger-Poisson systems involving multiple competing potentials
论文作者
论文摘要
在本文中，考虑了涉及多个竞争潜力和关键Sobolev指数的Schrödinger-Poisson系统。这样的问题不能以非线性术语的相同参数为正面的潜力来研究，因为重量电位设置$ \ {q_i（x）| 1 \ le i \ le m \} $包含非阳性，签名变化和非止痛元素。通过引入地面能量函数和微妙的分析，我们首先证明了基态解决方案的存在$ v_ \ varepsilon $在半经典限制中通过Nehari歧管和浓度 - 触觉原理。然后，我们表明$ v_ \ varepsilon $收敛到相关限制问题的基态解决方案，并在以电势为特征的混凝土集中浓缩。同时，还研究了基态溶液的某些特性。此外，获得基态解决方案不存在的足够条件。
In this paper, a class of Schrödinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set $\{Q_i(x)|1\le i \le m\}$ contains nonpositive, sign-changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution $v_\varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_\varepsilon$ converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.
