arXiv subjects

Guozhen Lu

Publications and source records attributed to Guozhen Lu.

At least 55 records · Page 3Linked to original sources

arXiv2023

$L^{p}$-Caffarelli-Kohn-Nirenberg inequalities and their stabilities

We establish a general identity (Theorem 1.2) that implies both the $L^{p}$-Hardy identities and the $L^{p}$-Caffarelli-Kohn-Nirenberg identities (Theorems 1.3 and 1.4) and $L^{p}$-Hardy inequalities and the $L^{p}$-Caffarelli-Kohn-Nirenberg inequalities (Theorems 1.5 and 1.6)). Weighted $L^{p}$-Caffarelli-Kohn-Nirenberg inequalities with nonradial weights are also obtained. (Theorem 1.7). Our results provide simple interpretations to the sharp constants, as well as the existence and non-existence of the optimizers, of several $L^{p}$-Hardy and $L^{p}% $-Caffarelli-Kohn-Nirenberg inequalities. As applications of our main results, we are able to establish stabilities of a class of $L^{2}$ and $L^{p}% $-Caffarelli-Kohn-Nirenberg inequalities. (Theorems 1.8 and 1.9.) We also derive the best constants and explicit extremal functions for a large family of $L^{2}$ and $L^{p}$ Caffarelli-Kohn-Nirenberg inequalities. (Corollaries 1.1 and 1.2.)

math.AP↗

arXiv2023

Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces

This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory.

math.AP↗

arXiv2023

A sharp trace Adams' inequality in $\mathbb{R}^{4}$ and Existence of the extremals

Let $Ω\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partialΩ$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(Ω)$ with zero mean value and zero Neumann boundary condition: \begin{equation*} S(α)=\underset{\int_Ωudx=0,\frac{\partial u}{\partialν}|_{\partialΩ}=0,\VertΔu\Vert_{2}\leq{1}}{\underset {u\in{W^{2,2}(Ω)\setminus\{0\}}}{\sup}}\int_{\partial Ω} e^{αu^{2}}dσ<\infty \end{equation*} holds if and only if $ α\leq12π^2$. Moreover, we prove a classification theorem for the solutions of a class of nonlinear boundary value problem of bi-harmonic equations on the half space $\mathbb{R}^4_{+}$. With this classification result, we can show that $S({12π^2})$ is attained by using the blow-up analysis and capacitary estimate. As an application, we prove a sharp trace Adams-Onofri type inequality in general four dimensional bounded domains with smooth boundary.

math.AP↗

arXiv2023

A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators

In this paper, we investigate the Hörmander type theorems for the multi-linear and multi-parameter Fourier multipliers. When the multipliers are characterized by $L^u$-based Sobolev norms for $1<u\le 2$ , our results on the smoothness assumptions are sharp in the multi-parameter and bilinear case. In the multi-parameter and multi-linear case, our results are almost sharp. Moreover, even in the one-parameter and multi-linear case, our results improve earlier ones in the literature.

math.CA↗

arXiv2023

Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains

We establish embeddings on a class of Sobolev spaces with potential weights on unbounded domains. Our results provide embeddings into weighted Lebesgue spaces $L^q_θ$ with radial power weights and establish the existence and non-existence of the maximizers for their Trudinger-Moser type inequalities. We also sharpen the maximal integrability by ``removing" necessary terms from the exponential series while maintaining the continuity of the embedding.

math.AP↗

arXiv2023

Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth

In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger-Moser exponential growth: \begin{equation*}\begin{cases} -dΔu_d+u_d=u_d(e^{u^2_{d}}-1)\ \ \mbox{in}\ Ω,\\ \frac{\partial u_d}{\partialν}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial Ω,\\ \end{cases}\end{equation*} where $d>0$ is a parameter, $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $ν$ is the unit outer normal to $\partial Ω$. We first prove the existence of a ground state solution to the above equation. If $d$ is sufficiently small, we prove that any ground state solution $u_d$ has at most one maximum point which is located on the boundary of $Ω$ and characterize the shape of ground state solution $u_d$ around the condensation point $P_d$. The key point of the proof lies in proving that the maximum point $P_d$ is close to the boundary at the speed of $\sqrt{d}$ when $d\rightarrow0$ and $u_d$ under suitable scaling transform converges strongly to the ground state solution of the limit equation $Δw+w=w(e^{w^2}-1)$. Our proof is based on the energy threshold of cut-off function, the concentration compactness principle for the Trudinger-Moser inequality, regularity theory for elliptic equation and an accurate analysis for the energy of the ground state solution $u_d$ as $d\rightarrow0$. Furthermore, by assuming that $Ω$ is a unit disk, we remove the smallness assumption on $d$ and show the maximum point of ground state solution $u_d$ must lie on the boundary of $Ω$ for any $d>0$.

math.AP↗

arXiv2023

Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

math.AP↗

arXiv2022

Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Uncertainty Principle and several stability results of the Caffarelli-Kohn-Nirenberg inequalities.

math.AP↗

arXiv2022

Uniqueness of positive solutions to elliptic equations with the critical exponential growth on the unit disc and its applications

In this paper, we will solve this uniqueness problem of positive solutions to the following equations of exponential growth: \begin{equation*} \begin{cases} -Δu =λue^{u^2},\quad\quad & x\in B_1\subset \mathbb{R}^2,\\ u>0, & x\in B_1,\ \\ u=0,\quad\quad &x\in \partial B_1, \end{cases} \end{equation*} where $ 0<λ<λ_1(B_1)$ and $λ_1(B_1)$ denotes the first eigenvalue of the operator $-Δ$ with the Dirichlet boundary in unit disk. Our method relies on delicate and difficult analysis of radial solutions to the above equation and careful asymptotic expansion of solutions near the boundary. This uniqueness result will shed some light on solving the conjecture that maximizers of the Trudinger-Moser inequality on the unit disc are unique. Furthermore, based on this uniqueness result, we develop a new strategy to establish the quantization property of elliptic equations with the critical exponential growth in the balls of hyperbolic spaces, and obtain the multiplicity and non-existence of positive critical points for super-critical Trudinger-Moser functional. Our method for the quantization property and non-existence of the critical points avoids using the complicated blow-up analysis used in the literature. This method can also be applied to study the similar problems in balls of high dimensional Euclidean space $\mathbb{R}^n$ or hyperbolic spaces provided the uniqueness for the corresponding quasilinear elliptic equations with the critical exponential growth is established.

math.AP↗

arXiv2022

Sharp Stability of Log-Sobolev and Moser-Onofri inequalities on the Sphere

In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere $\mathbb{S}^n$. Namely, we will establish the stability for Beckner's log-Sobolev inequality and Beckner's Moser-Onofri inequality on the sphere. We also prove that the sharp constant of global stability for the log-Sobolev inequality on the sphere $\mathbb{S}^n$ must be strictly smaller than the sharp constant of local stability for the same inequality. Furthermore, we also derive the non-existence of the global stability for Moser-Onofri inequality on the sphere $\mathbb{S}^n$.

math.AP↗

arXiv2022

Symmetry of solutions to higher and fractional order semilinear equations on hyperbolic spaces

We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space $\mathbb{H}^n$. Applying the Helgason-Fourier analysis techniques on $\mathbb{H}^n$, we develop a moving plane approach for integral equations on $\mathbb{H}^n$. We also establish the symmetry to solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtain symmetry to solutions of some semilinear equations involving fractional order derivatives.

math.AP↗

arXiv2022

Extendability and the $\overline \partial$ Operator on the Hartogs Triangle

In this paper it is shown that the Hartogs triangle $\mathbf T$ in $\mathbf C^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $\mathbf T$.

math.CV↗

arXiv2021

The Chang-Marshall Trace Inequality for Sobolev functions in domains in higher dimensional space $\mathbb{R}^n$

In their celebrated work [5], Chang and Marshall established a critical trace inequality of Moser-Trudinger type for holomorphic functions with mean value zero on unit disc in the complex plane. The main purpose is to address a question proposed to us by S. Y. Alice Chang who asked whether the Chang-Marshall type inequality for holomorphic functions on unit disk in the complex plane holds for Sobolev functions on general domains in higher dimensional Euclidean space $\mathbb{R}^n$ for all $n\ge 2$. We partially answer her question affirmatively.

math.CV↗

arXiv2021

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

math.AP↗

arXiv2021

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 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.

math.AP↗

arXiv2021

Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $\mathbb{S}^n$

In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on $\mathbb{S}^n$. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in $\mathbb{S}^n$, we first employ the Mobius transform between $\mathbb{S}^n$ and $\mathbb{R}^n$, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on $\mathbb{S}^n$ is equivalent to some integral equation in $\mathbb{R}^n$. Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on $\mathbb{S}^n$. As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on $\mathbb{S}^n$. Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on $\mathbb{S}^n$.

math.AP↗

arXiv2021

Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

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-Maz'ya inequalities and careful study of delicate properties of Green's functions of $P_k-λ$ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Green's functions allow us to obtain the integral representations of solutions and thus to avoid using the maximum principle to establish the symmetry of solutions.

math.AP↗

arXiv2021

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

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 Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz'ya 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 Geller's operators" and ``Octonionic Geller's 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 Poincaré-Sobolev and Hardy-Sobolev-Maz'ya 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 Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type.

math.CA↗