具有临界指数增长性的双调和方程的基态
Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials
摘要：
本文首先给出了一类非线性泛函在$H^{2}(\mathbb{R}^4)$中有界和紧的充要条件。利用这个结果和对称临界性原理，我们可以给出半线性双调和方程\[ (-Δ)^{2}u+γu=f(u)\ \text{in}\ \mathbb{R}^4 \]非平凡解的存在性与$γ\in \mathbb{R}^{+}$的值域之间的关系，其中$f(s)$是在无穷远处具有临界指数增长性的一般非线性项.本文的下一个目标是利用Fourier重排和Pohozaev恒等式证明当$V(x)$为正常数时方程\begin{equation}\label{con} (-Δ)^{2}u+V(x)u=λs\exp(2|s|^{2}))\ \text{in}\ \mathbb{R}^{4}, \end{equation}基态解的存在性.当$V(x)$是Rabinowitz型陷阱势时，即满足$$0<V_{0}=\underset{x\in\mathbb{R}^{4}}{\inf}V(x) <\underset{\ | x\ | \rightarrow\infty}{\lim}V(x) < +\infty. $$时，我们将探讨Nehari流形与相应的极限Nehari流形之间的关系，从而得到上述方程基态解的存在性。同样的结果和证明也适用于$\mathbb{R}^2$中含有Rabinowitz型陷阱势的具有临界指数增长性的调和方程。
英文摘要：
In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness for a class of nonlinear functionals in $H^{2}(\mathbb{R}^4)$. Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form \[ (-Δ)^{2}u+γu=f(u)\ \text{in}\ \mathbb{R}^4 \] and the range of $γ\in \mathbb{R}^{+}$, where $f(s)$ is the general nonlinear term having the critical exponential growth at infinity. Our next goal in this paper is to establish the existence of the ground-state solutions for the equation \begin{equation}\label{con} (-Δ)^{2}u+V(x)u=λs\exp(2|s|^{2}))\ \text{in}\ \mathbb{R}^{4}, \end{equation} when $V(x)$ is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the above equation when $V(x)$ is the Rabinowitz type trapping potential, namely it satisfies $$0<V_{0}=\underset{x\in\mathbb{R}^{4}}{\inf}V(x) <\underset{\ | x\ | \rightarrow\infty}{\lim}V(x) < +\infty. $$ The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $\mathbb{R}^2$.
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