双调和方程基态解的存在性

Existence of Ground-Sate Solutions for Biharmonic Equations

摘要: 文章主要研究了如下双调和方程$\begin{array}{c}{\Delta }^{2}u-\alpha \Delta u+u=Q(x)|u{|}^{p-2}u\text{, }x\in {ℝ}^N\end{array}$ ，其中$N\ge 5,2<2^{\ast \ast }=\frac{2N}{N-4},\alpha \in \left(-2,+\infty \right)$ 是一个常数，位势函数$Q\left(x\right):{ℝ}^N\to ℝ$ 是一个正连续函数。在位势函数$Q$ 满足合适的假设下，通过应用集中紧性引理和比较讨论的方法得到上述方程基态解的存在性。

Abstract: In this paper, consider the following biharmonic equations $\begin{array}{c}{\Delta }^{2}u-\alpha \Delta u+u=Q(x)|u{|}^{p-2}u\text{, }x\in {ℝ}^N\end{array}$ , where $N\ge 5,2<2^{\ast \ast }=\frac{2N}{N-4},\alpha \in \left(-2,+\infty \right)$ is a constant and the potential function $Q\left(x\right):{ℝ}^N\to ℝ$ is a positive continuous function. Under the appropriate assumption of the potential function Q, by applying the concentration compactness lemma and comparative discussion methods to obtain the existence of the ground-state solutions for the above equation.