AI 中文总结
本文针对带对数非线性的平面Schrodinger-Poisson系统，利用辅助泛函的极小极大原理证明非平凡解存在，并通过能量极小化得到基态解，在径向对称势下还获得无穷多个非径向变号解。
AI 中文摘要
本文研究如下平面Schrodinger-Poisson系统：$\begin{cases} -\Delta u +V(x)u+\lambda\phi u=u\ln u^2, & \text{in }{R}^2,\\\\ \Delta\phi=u^2, & \text{in }{R}^2. \end{cases}$ 其中$\lambda\in{R}$是一个参数，$V\in C^1({R}^2,R}^{+})$是一个强制势。由于对数非线性的存在，Ruiz [R] 发展的流形方法在此不适用。我们将一个一般的极小极大原理应用于辅助泛函，证明了非平凡解的存在性。然后，通过在所有非平凡解的集合上极小化能量泛函，我们成功地证明了该系统存在一个基态解。此外，如果$V$是径向对称的，我们获得无穷多个非径向变号解。
英文摘要
This paper is concerned with the following planar Schrodinger-Poisson system: $\begin{cases} -Δu +V(x)u+λϕu=u\ln u^2, & \text{in }{R}^2,\\ Δϕ=u^2, & \text{in }{R}^2. \end{cases}$ where $λ\in{R}$ is a parameter and $V\in C^1({R}^2,R}^{+})$ is a coercive potential. Due to the presence of the logarithmic nonlinearity, the manifold method developed by Ruiz [R] is not applicable here. We apply a general minimax principle to an auxiliary functional and prove the existence of a nontrivial solution. Then, by minimizing the energy functional over the set of all nontrivial solutions, we succeed in showing that the system admits a ground state solution. If, in addition, $V$ is radially symmetric, we obtain infinitely many non radial sign-changing solutions.
