arXiv subjects

Maochun Zhu

Publications and source records attributed to Maochun Zhu.

16 recordsLinked to original sources

arXiv2026

A complete characterization of the existence of extremals for the Trudinger-Moser inequality on $\mathbb{R}^2$ under sharp $L^p$-perturbations

In this paper, we investigate the following critical Trudinger--Moser inequality on $\mathbb R^2$ under sharp $L^p$-perturbations: $$ S(λ,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4πu^2}-1-λ|u|^p\right)\,dx . $$ For $2 λ^{\ast}$. Moreover, we show that the nonattainment in this range is caused by a vanishing phenomenon. For $p=2$, combining our analysis with the nonexistence results for $L^2$-perturbed Trudinger--Moser inequalities obtained in \cite{Chenluzhu}, we establish the existence of two finite thresholds $λ_{\ast}>-\infty$ and $λ^{\ast}<+\infty$ such that $S(λ,2)$ is attained when $λ_{\ast}<λ<λ^{\ast}$, and is not attained when $λ<λ_{\ast}$ or $λ>λ^{\ast}$. In contrast, for $p>4$, we prove that $S(λ,p)$ is attained for all admissible values of $λ$. Our results indicate that, in the whole-space setting, the $L^p$-perturbation term affects the existence and nonexistence of extremals through either concentration or vanishing phenomena, which is fundamentally different from the bounded-domain case, where existence or nonexistence is governed solely by concentration phenomena. These results provide a complete characterization of how sharp $L^p$ perturbations determine the existence and nonexistence of extremals for critical Trudinger--Moser inequalities on the entire $\mathbb R^2$. The resulting existence and nonexistence theory exhibits a threshold structure with respect to the $L^p$ pertubation reminiscent of the classical Brezis--Nirenberg phenomenon in the whole space $\mathbb R^2$.

math.AP↗

arXiv2026

Classification of solutions to the singular Liouville's equation associated with the $N$ Finsler Laplacian

In this paper, we classify a class of singular Liouville's equation associated with the Finsler-$N$-Laplacian for any $β\in (0,N)$ \begin{align*} -\mathrm{div}\left(F^{N-1}(\nabla u)DF(\nabla u)\right)=\hat{F}^{o}(x)^{-β}e^u\ \ \text{in } \mathbb{R}^{N}\backslash \{0\}, \end{align*} under the finite mass condition $\int_{\mathbb{R}^{N}}\hat{F}^{o}(x)^{-β}e^u dx<+\infty$. Here $F$ is a convex function, which is positively homogeneous of degree 1, and its polar $F^{o}$ represents a Finsler metric on $\mathbb{R}^{N}$, $\hat{F}^{o}(x)=F^{o}(-x)$. Our result relaxes the mass condition required in the classification result in [39]

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arXiv2025

Existence and Nonexistence of Extremals for Trudinger-Moser inequalities with $L^p$ type perturbation on any bounded planar domains

In this study, we investigate the perturbed Trudinger-Moser inequalities as follows:\[ S_Ω(λ,p)=\sup_{u\in H_{0}^{1}(Ω),\Vert\nabla u\Vert _{L^{2}\left( Ω\right) }\leq 1}\int_Ω\left( e^{4πu^{2}}-λ|u|^{p}\right) dx, \] where $1\leq p<\infty$ and $Ω$ is a bounded domain in $\mathbb{R}^2$. Our results demonstrate that there exists a threshold $λ^{\ast}(p)>0$ such that $S_Ω(λ,p)$ is attainable if $λ<λ^{\ast}(p)$, but unattainable if $λ>λ^{\ast}(p)$ when $p\in[1,2]$. For $p>2$, however, we show that $S_Ω(λ,p)$ is always attainable for any $λ\in \mathbb{R}$. These results are achieved through a refined blow-up analysis, which allow us to establish a sharp Dirichlet energy expansion formula for sequences of solutions to the corresponding Euler-Lagrange equations. The asymmetric nature of our problem poses significant challenges to our analysis. To address these, we will establish an appropriate comparison principle between radial and non-radial solutions of the associated Euler-Lagrange equations. Our study establishes a complete characterization of how $L^p$-type perturbations influence the existence of extremals for critical Trudinger-Moser inequalities on any bounded planar domains, this extends the classical Brezis-Nirenberg problem framework to the two-dimensional settings.

math.AP↗

arXiv2024

Optimal concentration level of anisotropic Trudinger-Moser functionals on any bounded domain

Let $F$ be convex and homogeneous of degree $1$, its polar $F^{o}$ represent a finsler metric on $\mathbb{R}^{n}$, and $Ω$ be any bounded open set in $\mathbb{R}^{n}$. In this paper, we first construct the theoretical structure of anisotropic harmonic transplantation. Using the anisotropic harmonic transplantation, co-area formula, limiting Sobolev approximation method, delicate estimate of level set of Green function, we investigate the optimal concentration level of the Trudinger-Moser functional \[ \int_Ωe^{λ_{n}|u|^{\frac{n}{n-1}}}dx \] under the anisotropic Dirichlet norm constraint $\int_ΩF^{n}\left( \nabla{u}\right) dx\leq1$, where $λ_{n}=n^{\frac{n}{n-1}}κ_{n}^{\frac{1}{n-1}}\ $ denotes the sharp constant of anisotropic Trudinger-Moser inequality in bounded domain and $κ_{n}$ is the Lebesgue measure of the unit Wulff ball. As an application. we can immediately deduce the existence of extremals for anisotropic Trudinger-Moser inequality on bounded domain. Finally, we also consider the optimal concentration level of the anisotropic singular Trudinger-Moser functional. The method is based on the limiting Hardy-Sobolev approximation method and constructing a suitable normalized anisotropic concentrating sequence.

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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.

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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.

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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.

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arXiv2019

Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials

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 $\mathbb{R}^2$.

math.AP↗

arXiv2019

Existence and Nonexistence of Extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2

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 $Ω\subset \mathbb{R}^n$, 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 Pólya-Szegö\ 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 $$ S(α)=\sup_{\|u\|_{H^2}=1}\int_{\mathbb{R}^4}\big(\exp(32π^2|u|^2)-1-α|u|^2\big)dx,$$ where $α\in (-\infty, 32π^2)$. We establish the existence of the threshold $α^{\ast}$, where $α^{\ast}\geq \frac{(32π^{2})^2B_{2}}{2}$ and $B_2\geq \frac{1}{24π^2}$, such that $S\left( α\right) $ is attained if $32π^{2}-α<α^{\ast}$, and is not attained if $32π^{2}-α>α^{\ast}$. 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.

math.AP↗

arXiv2017

Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg Groups and existence of ground state solutions

Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group $\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\left( \mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$. Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $\mathbb{H}^n$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $\mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth $\exp\left( αt^{\frac{Q}{Q-1}}\right) $ as $t\rightarrow+\infty$.

math.AP↗

arXiv2017

A sharp Trudinger-Moser type inequality involving $L^{n}$ norm in the entire space $\mathbb{R}^{n}$

Let $W^{1,n} ( \mathbb{R}^{n} $ be the standard Sobolev space and $\left\Vert \cdot\right\Vert _{n}$ be the $L^{n}$ norm on $\mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm \[ \underset{\left\Vert u\right\Vert _{W^{1,n}\left(\mathbb{R} ^{n}\right) }=1}{\sup}\int_{ \mathbb{R}^{n}}Φ\left( α_{n}\left\vert u\right\vert ^{\frac{n}{n-1}}\left( 1+α\left\Vert u\right\Vert _{n}^{n}\right) ^{\frac{1}{n-1}}\right) dx<+\infty \]in the entire space $\mathbb{R}^n$ for any $0\leqα<1$, where $Φ\left( t\right) =e^{t}-\underset{j=0}{\overset{n-2}{\sum}}% \frac{t^{j}}{j!}$, $α_{n}=nω_{n-1}^{\frac{1}{n-1}}$ and $ω_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $\mathbb{R}^n$. We also show that the above supremum is infinity for all $α\geq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $α>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work \cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $Φ(t)$ is replaced by a strictly smaller $Φ^*(t)=e^{t}-\underset{j=0}{\overset{n-1}{\sum}}% \frac{t^{j}}{j!}$. (Note that $Φ(t)=Φ^*(t)+\frac{t^{n-1}}{(n-1)!}$).

math.AP↗

arXiv2012

Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued bounded measurable functions defined in a bounded domain A of R^n, satisfying the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined by the Carnot-Caratheodory distance induced by the X_i's). We prove interior HW^{1,p} estimates (2<p<\infty) for weak solutions to the system.

math.AP↗

arXiv2011

Estimates in Generalized Morrey Spaces for Weak Solutions to Divergence Degenerate Parabolic Systems

Let $\mathrm{X}=(X_{1},...,X_{q})$ be a family of real smooth vector fields satisfying Hömander's condition. The purpose of this paper is to establish gradient estimates in generalized Morrey spaces for weak solutions of the divergence degenerate parabolic system related to $X$ :%\[u_{t}^{i}+X_α^{\ast}(a_{ij}^{αβ}(z)X_βu^{j}%)=g_{i}+X_α^{\ast}f_{i}^α(z), \] where $α,β=1,2,...,q,$ $i,j=1,2,...,N$, $X_α^{\ast}$ is the transposed vector field of $X_α$, $z=(t,x)\in{\mathbb{R}}^{n+1}$, and coefficients $a_{ij}^{αβ}(z)$ belong to the space $VMO$ induced by the vector fields $X_{1}, ...,X_{q}$.

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arXiv2011

L^p and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying Hörmander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are Hölder continuous with respect to the distance induced by the vector fields, then local Schauder estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space VMO with respect to the distance induced by the vector fields, then local L^{p} estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the presence of the drift term X_{0}, so that our class of operators covers, for instance, Kolmogorov-Fokker-Planck operators.

math.AP↗

arXiv2011

Local real analysis in locally homogeneous spaces

We introduce the concept of locally homogeneous space, and prove in this context L^p and Holder estimates for singular and fractional integrals, as well as L^p estimates on the commutator of a singular or fractional integral with a BMO or VMO function. These results are motivated by local a-priori estimates for subelliptic equations.

math.FA↗