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} (-Delta)^{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 $gamma^{*}$ such that for any $gammain (0,gamma^*)$, 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 $gammain (gamma^{*},+infty)$. 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.
In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and sc
aling invariance. These trace Trudinger-Moser and Adams inequalities can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders. Furthermore, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams inequalities.
The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators $P_k$ on bounded domains in the hyperbolic space $mathbb{
H}^n$ and as well as on the entire hyperbolic space $mathbb{H}^n$. Among other techniques, one of our main novelties is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Mazya inequalities and careful study of delicate properties of Greens functions of $P_k-lambda$ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Greens functions allow us to obtain the integral representations of solutions and thus to avoid using the maximum principle to establish the symmetry of solutions.
Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher
order Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on these spaces which are of their independent interests. To this end, we need to identify and introduce the ``Quaternionic Gellers operators and ``Octonionic Gellers operators which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, the precise heat and Bessel-Green-Riesz kernel estimates and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincare-Sobolev and Hardy-Sobolev-Mazya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient for us to establish, as a byproduct, the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works, completes our study of the factorization theorems, higher order Poincare-Sobolev, Hardy-Sobolev-Mazya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincare-Sobolev, Hardy-Sobolev-Mazya, Adams and Hardy-Adams inequalities on real hyperbolic
spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group $SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincare-Sobolev, Hardy-Sobolev-Mazya inequality on the Siegel domain $mathcal{U}^{n}$ and the unit ball $mathbb{B}_{mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. The factorization theorem we proved is of its independent interest in the Heisenberg group and CR sphere and CR invariant differential operators therein.
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 rel
ationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form [ (-Delta)^{2}u+gamma u=f(u) text{in} mathbb{R}^4 ] and the range of $gammain mathbb{R}^{+}$, 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} (-Delta)^{2}u+V(x)u=lambda sexp(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 $$0<V_{0}=underset{xinmathbb{R}^{4}}{inf}V(x) <underset{ | x | rightarrowinfty}{lim}V(x) < +infty. $$ The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $mathbb{R}^2$.
Using the Fourier analysis techniques on hyperbolic spaces and Greens function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $frac{n-1}{2}$-th order Hardy-Sobolev
-Mazya inequality in the upper half space of dimension $n$ coincides with the best $frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $ngeq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Mazya inequality in upper half space in the remaining cases of dimension $n$ and $k$-th order derivatives (see Theorem 1.7). Precise expressions and optimal bounds for Greens functions of the operator $ -Delta_{mathbb{H}}-frac{(n-1)^{2}}{4}$ on the hyperbolic space $mathbb{B}^n$ and operators of the product form are given, where $frac{(n-1)^{2}}{4}$ is the spectral gap for the Laplacian $-Delta_{mathbb{H}}$ on $mathbb{B}^n$. Finally, we give the precise expression and optimal pointwise bound of Greens function of the Paneitz and GJMS operators on hyperbolic space, which are of their independent interest (see Theorem 1.10).
Though much work has been done with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in $W^{1,n}(mathbb{R}^n)$ and higher order Adams inequalities on finite domain $Omegasubset mathbb{R}^n$, whether there
exists an extremal function for the critical higher order Adams inequalities on the entire space $mathbb{R}^n$ still remains open. The current paper represents the first attempt in this direction. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the P{o}lya-Szeg{o} type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in $mathbb{R}^4$ of the form $$ S(alpha)=sup_{|u|_{H^2}=1}int_{mathbb{R}^4}big(exp(32pi^2|u|^2)-1-alpha|u|^2big)dx,$$ where $alpha in (-infty, 32pi^2)$. We establish the existence of the threshold $alpha^{ast}$, where $alpha^{ast}geq frac{(32pi^{2})^2B_{2}}{2}$ and $B_2geq frac{1}{24pi^2}$, such that $Sleft( alpharight) $ is attained if $32pi^{2}-alpha<alpha^{ast}$, and is not attained if $32pi^{2}-alpha>alpha^{ast}$. This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on $mathbb{R}^2$. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Mazya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality
on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss work concerning the sharp constant in the Hardy-Sobolev-Mazya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Mazya inequality for bi-Laplacian in the upper half space of dimension five coincides with the Sobolev constant.
We establish sharp Hardy-Adams inequalities on hyperbolic space $mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $alpha>0$ there exists a constant $C_{alpha}>0$ such that [ int_{mathbb{B}^{4}}(e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2})d
V=16int_{mathbb{B}^{4}}frac{e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2}}{(1-|x|^{2})^{4}}dxleq C_{alpha}. ] for any $uin C^{infty}_{0}(mathbb{B}^{4})$ with [ int_{mathbb{B}^{4}}left(-Delta_{mathbb{H}}-frac{9}{4}right)(-Delta_{mathbb{H}}+alpha)ucdot udVleq1. ] As applications, we obtain a sharpened Adams inequality on hyperbolic space $mathbb{B}^{4}$ and an inequality which improves the classical Adams inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.
