Existence and non-existence of ground states of bi-harmonic equations involving constant and degenerate Rabinowitz potentials

Abstract

Recently, the authors of the current paper established in Chen et al. (Calc Var Partial Differ Equ 59:1–38, 2020) the existence of a ground-state solution to the following bi-harmonic equation with the constant potential or Rabinowitz potential:

$$\begin{aligned} (-\Delta )^{2}u+V(x)u=f(u)\ \text {in}\ \mathbb {R}^{4}, \end{aligned}$$

(0.1)

when the nonlinearity has the special form $f(t)=t(\exp (t^2)-1)$ and $V(x)\ge c>0$ is a constant or the Rabinowitz potential. One of the crucial elements used in Chen et al. (Calc Var Partial Differ Equ 59:1–38, 2020) 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 Eq. (0.1) 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 $\gamma ^{*}$ such that for any $\gamma \in (0,\gamma ^*)$, the Eq. (0.1) with the constant potential $V(x)=\gamma >0$ admits a ground-state solution, while does not admit any ground-state solution for any $\gamma \in (\gamma ^{*},+\infty )$. The second purpose of this paper is to establish the existence of a ground-state solution to the Eq. (0.1) 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.