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 [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 $V(x)\ge c>0$ 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 ${\gamma }^{\ast }$ such that for any $\gamma \in (0,{\gamma }^{\ast })$, the above equation with the constant potential $V(x)=\gamma >0$ admits a ground-state solution, while does not admit any ground-state solution for any $\gamma \in ({\gamma }^{\ast },+\infty )$. 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.

31 pages. arXiv admin note: text overlap with arXiv:1909.01952