论文标题
具有积极潜力的平面Schrödinger-Poisson系统
The planar Schrödinger-Poisson system with a positive potential
论文作者
论文摘要
在本文中，我们考虑问题\ begin {equination*} \ left {\ begin {array} {l} {l}-ΔU\ pm ϕu + w'（x，x，x，u）= 0 \ hbox {in} \ end {array} \ right。 \ end {equation*}，其中$ w $假定为正。在第三维中，符号 +（我们称其为$（\ Mathcal p _ +）$）的问题被考虑并在\ cite {m}中解决，而在同一篇论文中，如果我们考虑符号，则不存在非平凡的解决方案 - （假设它$（\ Mathcal p _--））$）。我们为$（\ Mathcal p _+）$提供了一个一般存在的结果，并且在情况下至少存在至少一个非平凡的解决方案的情况。
In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -Δu \pm ϕu + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline Δϕ= u^2 \hbox{ in } \mathbb{R}^2, \end{array} \right. \end{equation*} where $W$ is assumed positive. In dimension three, the problem with the sign + (we call it $(\mathcal P_+)$) was considered and solved in \cite{M}, whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign -- (say it $(\mathcal P_-)$). We provide a general existence result for $(\mathcal P_+)$ and two examples falling in the case $(\mathcal P_-)$ for which there exists at least a nontrivial solution.
