论文标题
平面Schrödinger-Poisson系统的组不变解决方案
Group invariant solutions for the planar Schrödinger-Poisson system
论文作者
论文摘要
本文与以下平面Schrödinger-Poisson System \ Begin {equation*} \ begin {case} - \三角{ \triangle往|通过假设非线性$ f（x，t）$具有关键的指数增长，我们可以为上述系统获得非平凡的解决方案或Nehari类型的基态解决方案。我们的结果扩展了CAO_DAI_ZHANG和CHEN-TANG的先前作品。我们处理更通用的非线性$ f $，而无穷大的约束弱，我们只假设（AR）类型的条件可以实现单调性假设。我们考虑了所有情况$ p \ geq2 $，并显示了具有多种对称性的解决方案的存在。就像在Chen_tang中一样，我们采用了山通道结构的版本，该结构提供了一个具有两个创新点的陶瓷序列。首先，我们对能量功能的关键部分的迹象进行关键观察，该迹象对应于非局部项$ ϕ | u | u |^{p-2} u $，其次，我们采用新的Moser型函数来确保Cerami序列的界限和紧凑性。此外，我们的方法也适用于亚临界增长案例，并概括了最近的作品liu_radulescu_tang_zhang，cao_dai_zhang，chen_tang。
This paper is concerned with the following planar Schrödinger-Poisson system \begin{equation*} \begin{cases} -\triangle{u}+V(x)u+ϕ{(x)}|u|^{p-2}u=f(x,u),&\text{in $\mathbb{R}^{2}$}, \triangleϕ=|u|^{p},&\text{in $\mathbb{R}^{2}$}, \end{cases} \end{equation*} where $p\geq2$ is a constant, $V(x)$ and $f(x,t)$ are continuous, mirror symmetric or rotationally periodic functions. By assuming that the nonlinearity $f(x,t)$ has critical exponential growth, we obtain a nontrivial solution or a ground state solution of Nehari type to the above system. Our results extend previous works of Cao_Dai_Zhang and Chen-Tang. We handle more general nonlinearities $f$ with weaken constraint at infinity, and we assume only the (AR) type condition to take place of the monotonicity assumption. We considered all the cases $p\geq2$, and we show the existence of solutions with multiple types of symmetry. As in Chen_Tang, we adopt a version of mountain pass structure which provides a Cerami sequence, with two innovative points. First, we make a key observation for the sign of a crucial part of the energy functional corresponding to the nonlocal term $ϕ|u|^{p-2}u$, and secondly we adopt a new Moser type functions to ensure the boundedness and compactness of the Cerami sequence. Moreover, our approach works also for the subcritical growth case, and generalizes recent works Liu_Radulescu_Tang_Zhang,Cao_Dai_Zhang,Chen_Tang.
