含有常数和退化Rabinowitz势的双调和方程基态的存在性和不存在性
Existence and Non-existence of Ground states of bi-harmonic equations involving constant and degenerate Rabinowitz potentials
摘要：
最近，本文作者在[9]中建立了具有常势或Rabinowitz势的下列双调和方程的基态解的存在性：当非线性具有特殊形式$f(t)=t(\exp(t^2)-1)$且$V(x)\geq c>0$是常数或Rabinowitz势时，\begin{equation} (-Δ)^{2}u+V(x)u=f(u)\ \text{in}\ \mathbb{R}^{4}, \end{equation}。[9]中使用的关键要素之一是傅立叶重排参数。但是，如果$f(t)$不是奇数函数，则此参数不适用。因此，即使当$V(x)$是正常数时，上述具有一般临界指数非线性$f(u)$的方程是否允许基态解仍然是开放的。本文的第一个目的是发展一种无傅立叶反射的方法来解决上述问题。更确切地说，我们将证明存在一个阈值$γ^{*}$，使得对于任何$γ\in (0,γ^*)$，上述方程与恒定的电位$V(x)=γ>0$承认一个基态解，而不承认任何$γ\in (γ^{*},+\infty)$的任何基态解。本文的第二个目的是建立上述方程的基态解的存在性，其中任何退化的Rabinowitz势$V$在某个有界开集上为零。在其他技术中，证明还依赖于一个关键的亚当斯不等式，涉及退化的潜力，这是它自己的利益。
英文摘要：
Recently, the authors of the current paper established in [9] the existence of a ground-state solution to the following bi-harmonic equation with the constant potential or Rabinowitz potential: \begin{equation} (-Δ)^{2}u+V(x)u=f(u)\ \text{in}\ \mathbb{R}^{4}, \end{equation} when the nonlinearity has the special form $f(t)=t(\exp(t^2)-1)$ and $V(x)\geq c>0$ is a constant or the Rabinowitz potential. One of the crucial elements used in [9] is the Fourier rearrangement argument. However, this argument is not applicable if $f(t)$ is not an odd function. Thus, it still remains open whether the above equation with the general critical exponential nonlinearity $f(u)$ admits a ground-state solution even when $V(x)$ is a positive constant. The first purpose of this paper is to develop a Fourier rearrangement-free approach to solve the above problem. More precisely, we will prove that there is a threshold $γ^{*}$ such that for any $γ\in (0,γ^*)$, the above equation with the constant potential $V(x)=γ>0$ admits a ground-state solution, while does not admit any ground-state solution for any $γ\in (γ^{*},+\infty)$. The second purpose of this paper is to establish the existence of a ground-state solution to the above equation with any degenerate Rabinowitz potential $V$ vanishing on some bounded open set. Among other techniques, the proof also relies on a critical Adams inequality involving the degenerate potential which is of its own interest.
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