Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-10-09 , DOI: 10.1007/s00526-020-01831-4 Lu Chen , Guozhen Lu , Maochun Zhu
In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in \(H^{2}\left( {\mathbb {R}}^{4}\right) \) which are of their independent interests. (See Theorems 2.1 and 2.2.) 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
$$\begin{aligned} (-\Delta )^{2}u+\gamma u=f(u)\ \text {in}\ {\mathbb {R}}^{4} \end{aligned}$$and the range of \(\gamma \in {\mathbb {R}}^{+}\), where \(f\left( s\right) \) is the general nonlinear term having the critical exponential growth at infinity. (See Theorem 2.7.) Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, it seems that nothing is known so far about the existence of the ground-state solutions for this class of equations involving the trapping potential introduced by Rabinowitz (Z Angew Math Phys 43:27–42, 1992). Since the trapping potential is not necessarily symmetric, classical radial method cannot be applied to solve this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation
$$\begin{aligned} (-\Delta )^{2}u+V(x)u=\lambda s\exp (2|s|^{2}))\ \text {in}\ {\mathbb {R}}^{4}, \end{aligned}$$(0.1)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 Eq. (2.5) when V(x) is the Rabinowitz type trapping potential, namely it satisfies
$$\begin{aligned} 0<\inf _{x \in {\mathbb {R}}^{4}} V(x)<\sup _{x \in {\mathbb {R}}^{4}} V(x)=\lim _{|x| \rightarrow +\infty } V(x). \end{aligned}$$(See Theorem 2.8.) The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in \({\mathbb {R}}^2\). (See Theorem 2.9.)
中文翻译：
具有临界和临界势的临界指数增长的双调和方程的基态
在本文中，我们首先给出\（H ^ {2} \ left（{\ mathbb {R}} ^ {4} \ right）\中的一类非线性泛函的有界性和紧致性的充要条件），这是他们的独立利益。（请参见定理2.1和2.2。）使用该结果和对称临界原理，我们可以表示形式为半线性双调和方程的非平凡解的存在之间的关系
$$ \ begin {aligned}（-\ Delta）^ {2} u + \ gamma u = f（u）\ \ text {in} \ {\ mathbb {R}} ^ {4} \ end {aligned} $$以及\（\ gamma \ in {\ mathbb {R}} ^ {+} \）的范围，其中\（f \ left（s \ right）\）是在无穷大处具有临界指数增长的一般非线性项。（见定理2.7。）尽管在文献中已经研究了具有临界指数增长的双调和方程非平凡解的存在，但迄今为止，关于此的基态解的存在似乎一无所知Rabinowitz引入的涉及捕获势的一类方程（Z Angew Math Phys 43：27-42，1992）。由于捕获势不一定是对称的，因此经典的径向方法不能用于解决该问题。为了克服这个困难，我们首先建立方程的基态解的存在
$$ \ begin {aligned}（-\ Delta）^ {2} u + V（x）u = \ lambda s \ exp（2 | s | ^ {2}））\ \ text {in} \ {\ mathbb {R}} ^ {4}，\ end {aligned} $$（0.1）当使用傅立叶重排和Pohozaev等式时，V（x）是一个正常数。然后，我们将探索Nehari流形与相应的极限Nehari流形之间的关系，以得出等式的基态解的存在。（2.5）当V（x）是Rabinowitz型俘获势时，即满足
$$ \ begin {aligned} 0 <\ inf _ {x \ in {\ mathbb {R}} ^ {4}} V（x）<\ sup _ {x \ in {\ mathbb {R}} ^ {4 }} V（x）= \ lim _ {| x | \ rightarrow + \ infty} V（x）。\ end {aligned} $$（请参见定理2.8。）相同的结果和证明适用于具有临界指数增长且涉及\（{\ mathbb {R}} ^ 2 \）中的Rabinowitz型俘获势的调和方程。（请参见定理2.9。）
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