STANDING WAVES FOR NONLINEAR SCHRODINGER-POISSON EQUATION WITH HIGH FREQUENCY
STANDING WAVES FOR NONLINEAR SCHRODINGER-POISSON EQUATION WITH HIGH FREQUENCY
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DOI：
10.12775/tmna.2015.028
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发表时间：
2015-06
影响因子：
0.7
通讯作者：
Jianqing Chen;Zhengping Wang;Xiaoju Zhang
中科院分区：
文献类型：
--
作者：
Jianqing Chen;Zhengping Wang;Xiaoju Zhang
本文研究了如下Schrodinger-Poisson方程的基态和束缚态的存在性，其中$p\in（3，5）$，$\lambda > 0$，$V\in C（\mathbb{R}^3，\mathbb {R}^+）$和$\lim\limits_{|X|\to +\infty}V（x）=\infty$.利用变分方法证明了：对于任意$\lambda > 0$，存在$\delta_1（\lambda）> 0$使得对于$\mu_1 < \mu < \mu_1 + \delta_1（\lambda）$，问题（P）有一个从零解分支出来的具有负能量的非负基态;问题（P）存在一个正能量的非负束缚态，它不能从零解分叉。这里$\mu_1 $是$-\Delta +V$的第一个特征值。借助于对称山路定理的推广，我们得到了大量的非平凡束缚态。
We study the existence of ground state and bound state for the following Schrodinger-Poisson equation where $p\in(3,5)$, $\lambda > 0$, $V\in C(\mathbb{R}^3,\mathbb{R}^+)$ and $\lim\limits_{|x|\to +\infty}V(x)=\infty$. By using variational method, we prove that for any $\lambda > 0$, there exists $\delta_1(\lambda) > 0$ such that for $\mu_1 < \mu < \mu_1 + \delta_1(\lambda)$, problem (P) has a nonnegative ground state with negative energy, which bifurcates from zero solution; problem (P) has a nonnegative bound state with positive energy, which can not bifurcate from zero solution. Here $\mu_1$ is the first eigenvalue of $-\Delta +V$. Infinitely many nontrivial bound states are also obtained with the help of a generalized version of symmetric mountain pass theorem.