Adams Inequality Research Articles

Weighted Second Order Adams Inequality in the Whole Space $$\mathbb {R}^{4}$$

Weighted Second Order Adams Inequality in the Whole Space $$\mathbb {R}^{4}$$

Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane

The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller 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, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.

Attainability of Second Order Adams Inequalities with Navier Boundary Conditions

Attainability of Second Order Adams Inequalities with Navier Boundary Conditions

Multiplicity results for some fourth-order elliptic equations with combined nonlinearities

<abstract><p>In this paper, we regard with some fourth-order elliptic boundary value problems involving subcritical polynomial growth and subcritical (critical) exponential growth. Some new existence and multiplicity results are established by using variational methods combined Adams inequality.</p></abstract>

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

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.

Equivalence of Subcritical and Critical Adams Inequalities in $W^{m,2}(\mathbb {R}^{2m})$ and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints

Equivalence of Subcritical and Critical Adams Inequalities in $W^{m,2}(\mathbb {R}^{2m})$ and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints

Existence and Non-existence of Extremals for Critical Adams Inequality in any Even Dimension

Existence and Non-existence of Extremals for Critical Adams Inequality in any Even Dimension

Estimate for concentration level of the Adams functional and extremals for Adams-type inequality

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in the higher order Sobolev space with homogeneous Navier boundary conditions WNm,nm(Ω), which in particular includes the classical Sobolev space W0m,nm(Ω), where Ω is a smooth bounded domain in Euclidean n-space. Secondly, based on the Concentration-compactness alternative due to Do Ó and Macedo, we prove the existence of extremals for the Adams inequality under Navier boundary conditions for second order derivatives at least for higher dimensions when Ω is an Euclidean ball.

Sharp Critical and Subcritical Trace Trudinger–Moser and Adams Inequalities on the Upper Half-Spaces

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 scaling invariance. These trace Trudinger–Moser (Theorems 1.1 and 1.2) and trace Adams inequalities (Theorems 1.4, 1.5, 1.10 and 1.11) can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders on half-spaces. Furthermore, as an application, 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 (Theorem 1.13). It is interesting to note that there are two types of trace Trudinger–Moser and trace Adams inequalities: critical and subcritical trace inequalities under different constraints. Moreover, trace Trudinger–Moser and trace Adams inequalities of exact growth also hold on half-spaces (Theorems 1.6, 1.8 and 1.12).

Adams inequalities with exact growth condition for Riesz-like potentials on [formula omitted

We derive sharp Adams inequalities with exact growth condition for the Riesz potential and for more general Riesz-like potentials on Rn. We also obtain Moser-Trudinger inequalities with exact growth condition for the fractional Laplacian, and for general homogeneous elliptic differential operators with constant coefficients.

Singular Hardy–Adams inequalities on hyperbolic spaces of any even dimension

The main purpose of this paper is to establish singular Hardy–Adams inequalities on hyperbolic space $\mathbb B^n$ of any even dimension. More precisely, we will prove that for any even integer $n\geq 4$, $\alpha \gt 0$ and $0 \lt \beta \lt n$, there exis

Hamiltonian elliptic systems with critical polynomial-exponential growth

In this work, we revisit a classical problem of Hamiltonian-type systems of second-order elliptic PDE in bounded domains. Recalling the famous critical hyperbola associated with this problem, we propose an approach where one of the nonlinearities has a polynomial growth over this hyperbola’s asymptotes, which allows the other one to have exponential growth. We establish solutions for these critical systems by using the reduction by inversion method and some new results for sharp Adams Inequality and Concentration–Compactness for high order N-Laplace operators.

Hardy–Adams Inequalities on ℍ2 × ℝ n-2

Abstract Let ℍ 2 {\mathbb{H}^{2}} be the hyperbolic space of dimension 2. Denote by M n = ℍ 2 × ℝ n - 2 {M^{n}=\mathbb{H}^{2}\times\mathbb{R}^{n-2}} the product manifold of ℍ 2 {\mathbb{H}^{2}} and ℝ n - 2 ( n ≥ 3 ) {\mathbb{R}^{n-2}(n\geq 3)} . In this paper we establish some sharp Hardy–Adams inequalities on M n {M^{n}} , though M n {M^{n}} is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré–Sobolev inequality on M n {M^{n}} coincides with the best Sobolev constant, which is of independent interest.

Sharp Green’s Function Estimates on Hadamard Manifolds and Adams Inequality

Abstract In this article, we establish estimates on Riesz-type kernels and prove the Adams-type inequality for $W^{k,p}(M)$ functions, where $(M,g)$ is an $n$-dimensional Hadamard manifold with sectional curvature bounded from below and above by a negative constant and $k$ is an integer satisfying $kp=n$.

Moser–Trudinger–Adams inequalities and related developments

In this paper, we review some of the developments on the Moser–Trudinger and Adams inequalities in both Euclidean space and on Riemannian manifolds. We will also describe some of the closely related problems.

Sharp Rellich–Sobolev inequalities and weighted Adams inequalities involving Hardy terms for bi-Laplacian

Motivated by a recent work of Dong, Lam and Lu concerning the sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities, we derive a sharp Rellich–Sobolev inequality on Rn with n≥5 and weighted Adams inequalities involving Hardy terms on R4. The procedure is based on a quasi-conformal mapping type transform and decomposition into spherical harmonics since the symmetrization arguments do not work in dealing with these inequalities.

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities $$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}dx<+\infty;$$ moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).

Orlicz-fractional maximal operators in Morrey and Orlicz–Morrey spaces

In 1995, Perez introduced $$B_{p}$$ -condition, which is necessary and sufficient condition for the boundedness of the Orlicz maximal operator on $$L^{p}$$ spaces. After, necessary and sufficient condition of the Hardy–Littlewood–Sobolev type inequality for Orlicz-fractional maximal operator is derived by Cruz-Uribe and Moen in 2013. In this paper, we investigate the boundedness of Orlicz maximal operator, Orlicz-fractional maximal operator and fractional integral operator in Morrey and Orlicz–Morrey spaces on the assumption that each Young function satisfies these conditions, respectively. In particular, one of the main results is based on the Adams inequality in the framework of Morrey spaces.

Existence and nonexistence of extremals for critical Adams inequalities in [formula omitted] and Trudinger-Moser inequalities in [formula omitted

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W1,n(Rn) and higher order Adams inequalities on finite domain Ω⊂Rn, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space Rn still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R4. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31]), 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 R4 of the formS(α)=sup‖u‖H2=1⁡∫R4(exp⁡(32π2|u|2)−1−α|u|2)dx, where α∈(−∞,32π2). We establish the existence of the threshold α⁎, where α⁎≥(32π2)2B22 and B2≥124π2, such that S(α) is attained if 32π2−α<α⁎, and is not attained if 32π2−α>α⁎. This phenomenon 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 R2. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem (−Δ)mu=h(x,u) in Ω,u=∂nu=⋯=∂nm−1u=0 on ∂Ω,where h is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger–Moser–Adams inequality, either when Ω is a ball or, provided an energy control on solutions is prescribed, when Ω is a smooth bounded domain. Our results are sharp within the class of distributional solutions. The analogous problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory.