arXiv subjects

Quôc Anh Ngô

Publications and source records attributed to Quôc Anh Ngô.

17 recordsLinked to original sources

arXiv2026

Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order Hénon equation via Bernstein's technique

In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate \[-Δu \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac σ2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounded, positive, $C^4$-solution $u$ to the the fourth-order Hénon equation \[Δ^2 u = |x|^σu^p \quad \text{in } \mathbf R^n\] with $n \geq 5$, $σ\geq 0 $ but implicitly near $0$, and $p>(n+4+2σ)/(n-4)$. Their argument relies on a sophisticated iteration argument in the fashion of the standard Moser proof. In this paper, we provide an alternative approach which relies on the Bernstein technique via the maximum principle. The heart of our argument is a suitable choice of an auxiliary function allowing us not only achieving the same pointwise estimate, which still holds even for $p = (n+4+2σ)/(n-4)$, but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any $σ\geq 0$. This estimate leads to an interesting consequence of the existence of conformal metric of $\mathbf R^n$ having positive scalar and $Q$ curvatures. We also show that the coefficient $2/(n-4)$ is sharp, hence providing the first answer to the question in the op. cit. paper. Our proof appears to be simpler and constructive.

math.AP↗

arXiv2024

A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\mathbf R^n$

We consider the biharmonic equation $Δ^2 u = u^α$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $α\leqslant 1$ with $n \geqslant 1$ or $α\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $α\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $α\leqslant 1$ with $n \geqslant 1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators.

math.AP↗

arXiv2023

A Liouville type result for fractional GJMS equations on higher dimensional spheres

Let $n$ be an integer and $s$ be a real number such that $n > 2s \geq 2$. Inspired by the perturbation approach initiated by F. Hang and P. Yang (\textit{Int. Math. Res. Not. IMRN}, 2020), we are interested in non-negative, smooth solution $v$ to the following higher-order fractional equation \[ {\mathbf P}_n^{2s}(v) = Q_n^{2s}(\varepsilon v+v^α) \] on $\mathbf S^n$ with $0<α\leq (n+2s)/(n-2s)$, and $\varepsilon \geq 0$. Here ${\mathbf P}_n^{2s}$ is the fractional GJMS type operator of order $2s$ on $\mathbf S^n$ and $Q_n^{2s} ={\mathbf P}_n^{2s}(1)$ is constant. We show that if $\varepsilon >0$ and $0<α\leq (n+2s)/(n-2s)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon=0$ but with $0<α< (n+2s)/(n-2s)$.As a by-product, with $0<α\leq (n+2s)/(n-2s)$, we compute the sharp constant of the subcritical/critical Sobolev inequalities \[ \int_{\mathbf S^n} v {\mathbf P}_n^{2s} (v) dμ_{g_{\mathbf S^n}} \geq \frac{Γ(n/2 + s)}{Γ(n/2 - s )} | \mathbf S^n|^\frac{α-1}{α+1} \Big( \int_{\mathbf S^n} v^{α+1} dμ_{g_{\mathbf S^n}} \Big)^\frac{2}{α+1}. \] for the GJMS operator ${\mathbf P}_n^{2s}$ on $\mathbf S^n$ and for all non-negative functions $v\in H^s(\mathbf S^n)$.

math.AP↗

arXiv2023

An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences

For $n > k \geq 0$, $λ>0$, and $p, r>1$, we establish the following optimal Hardy-Littlewood-Sobolev inequality \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}{ |x-y|^λ|y"|^β} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^{n-k})} \| g\| _{L^r(\mathbf R^n)} \] with $y = (y', y") \in \mathbf R^{n-k} \times \mathbf R^k$ under the two necessary conditions \[ β< \left\{ \begin{aligned} & k - k/r & & \text{if } \; 0 < λ\leq n-k,\\ & n - λ- k/r & & \text{if } \; n-k < λ, \end{aligned} \right. \] and \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { β+ λ} n = 2 -\frac kn. \] We call this the optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$. The existence of an optimal pair for this new inequality is also studied. The motivation of working on the above inequality is to provide a unification of many known Hardy-Littewood-Sobolev inequalities including the classical Hardy-Littewood-Sobolev inequality when $k=β=0$, the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ when $k=1$ and $β= 0$, and the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ with extended kernel when $k=1$ and $β\ne 0$. We show that the above condition for $β$ is sharp. In the unweighted case, namely $β=0$, our finding immediately leads to the sharp Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ with the optimal range $$0<λ<n-k/r,$$ which has not been observed before, even in the case $k=1$. As one of many consequences, we give a short proof of the Stein-Weiss inequality in the context of $\mathbf R^{n-k} \times \mathbf R^n$.

math.FA↗

arXiv2023

On the Hang-Yang conjecture for GJMS equations on $\mathbb S^n$

This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation \[ {\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-α} ) \] on $\mathbb S^n$ with $m \geq 2$, $3 \leq n < 2m $, $0<α\leq (2m+n)/(2m-n)$, and $\varepsilon >0$. Here $ {\mathbf P}^{2m}_n$ is the GJMS operator of order $2m$ on $\mathbb S^n$ and $Q_n^{2m} =(2/(n-2m)) {\mathbf P}^{2m}_n (1)$ is constant. We show that if $\varepsilon >0$ is small and $0<α\leq (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon =0$ and $0<α< (2m+n)/(2m-n)$. In the special case $n=3$, $m=2$, and $α=7$, such Liouville type result was recently conjectured by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020). As a by-product, we obtain the sharp (subcritical and critical) Sobolev inequalities \[ \Big( \int_{\mathbb S^n} v^{1-α} dμ_{\mathbb S^n} \Big)^{\frac {2}{α-1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) dμ_{\mathbb S^n} \geq \frac{Γ(n/2 + m)}{Γ(n/2 - m )} | \mathbb S^n|^\frac{α+ 1}{α- 1} \] for the GJMS operator $ {\mathbf P}^{2m}_n$ on $\mathbb S^n$ under the conditions $n \geq 3$, $n=2m-1$, and $α\in(0,1) \cup (1, 2n+1]$. A log-Sobolev type inequality, as the limiting case $α=1$, is also presented.

math.AP↗

arXiv2022

Exhaustive existence and non-existence results for Hardy-Hénon equations in $\mathbf R^n$

This paper concerns solutions to the Hardy-Hénon equation \[ -Δu = |x|^σu^p \] in $\mathbf R ^n$ with $n \geq 1$ and arbitrary $p, σ\in \mathbf R$. This equation was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions $p>1$, $σ\geq -2$, and $n \geq 3$ is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.

math.AP↗

arXiv2021

All conditions for Stein-Weiss inequalities are necessary

The famous Stein-Weiss inequality on $\mathbf R^n \times \mathbf R^n$, also known as the doubly weighted Hardy-Littlewood-Sobolev inequality, asserts that \[ \Big| \iint_{\mathbf R^n \times \mathbf R^n} \frac{f(x) g(y)}{|x|^α|x-y|^λ|y|^β} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^n)} \| g\| _{L^r(\mathbf R^n)} \] holds for any $f\in L^p(\mathbf R^n)$ and $g\in L^r(\mathbf R^n)$ under several conditions on the parameters $n$, $p$, $r$, $α$, $β$, and $λ$. Extending the above inequality to either different domains rather than $\mathbf R^n \times \mathbf R^n$ or classes of more general kernels rather than the classical singular kernel $|x-y|^{-λ}$ has been the subject of intensive studies over the last three decades. For example, Stein-Weiss inequalities on the upper half space, on the Heisenberg group, on homogeneous Lie group are known. Served as the first step, this work belongs to a set in which the following inequality on the product $\mathbf R^{n-k} \times \mathbf R^n$ is studied \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}{|x|^α|x-y|^λ|y|^β} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^{n-k})} \| g\| _{L^r(\mathbf R^n)}. \] Toward the validity of the above new inequality, in this work, by constructing suitable counter-examples, we establish all conditions for the parameters $n$, $p$, $r$, $α$, $β$, and $λ$ necessarily for the validity of the above proposed inequality. Surprisingly, these necessary conditions applied to the case $k=1$ suggest that the existing Stein-Weiss inequalities on the upper half space are yet in the optimal range of the parameter $λ$. This could reflect limitations of the methods often used. Comments on the Stein-Weiss inequality on homogeneous Lie groups as well as the reverse form for Stein-Weiss inequalities are also made.

math.FA↗

arXiv2020

Existence and non-existence results for the higher order Hardy-Hénon equation revisited

This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions to the higher order Hardy-Hénon equations \[ (-Δ)^m u = |x|^σu^p \] in $\mathbf R^n$ with $p > 1$. We show that the condition \[ n - 2m - \frac{2m+σ}{p-1} >0 \] is necessary for the existence of distributional solutions. For $n \geq 2m$ and $σ> -2m$, we prove that any distributional solution satisfies an integral equation and a weak super polyharmonic property. We establish some sufficient conditions for punctured or classical solution to be a distributional solution. As application, we show that if $n \geq 2m$ and $σ> -2m$, there is no non-negative, non-trivial, classical solution to the equation if \[ 1 < p < \frac{n+2m+2σ}{n-2m}. \] At last, we prove that for for $n > 2m$, $σ> -2m$ and $$p \geq \frac{n+2m+2σ}{n-2m},$$ there exist positive, radially symmetric, classical solutions to the equation.

math.AP↗

arXiv2019

Prescribed $Q$-curvature flow on closed manifolds of even dimension

On a closed Riemannian manifold $(M,g_0)$ of even dimension $n \geqslant 4$, the well-known prescribed $Q$-curvature problem asks whether or not there is a metric $g$ comformal to $g_0$ such that its $Q$-curvature, associated with the GJMS operator $\mathbf P_g$, is equal to a given function $f$. Letting $g = e^{2u}g_0$, this problem is equivalent to solving \[ \mathbf P_{g_0} u+Q_{g_0} = f e^{nu}, \] where $Q_{g_0}$ denotes the $Q$-curvature of $g_0$. The primary objective of the paper is to introduce the following negative gradient flow of the time dependent metric $g(t)$ conformal to $g_0$, \[ \frac{\partial g (t)}{\partial t}= -2\Big(Q_{g (t)} - \frac{\int_M f Q_{g(t)} dμ_{g(t)} }{\int_M f^2 dμ_{g(t)} }f \Big)g(t) \quad \text{ for } t >0, \] to study the problem of prescribing $Q$-curvature. Since $\int_M Q_g dμ_g$ is conformally invariant, our analysis depends on the size of $\int_M Q_{g_0} dμ_{g_0}$, which is assumed to satisfy \[ \int_M Q_0 dμ_{g_0} \ne k (n-1)! \, {\rm vol}(\mathbb S^n) \quad \text{ for all } \; k = 2,3,... \] The paper is twofold. First, we identify suitable conditions on $f$ such that the gradient flow defined as above is defined to all time and convergent, as time goes to infinity, sequentially or uniformly. Second, we show that various existence theorems for prescribed $Q$-curvature problem can be derived from the convergence of the flow.

math.AP↗

arXiv2019

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ Δ^m u = \pm u^α \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $α\in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $α$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $α$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.

math.AP↗

arXiv2019

Supercritical Moser-Trudinger inequalities and related elliptic problems

Given $α>0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (α_n + |x|^α) |u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( α_n |u|^{\frac{n}{n-1} + |x|^α} \big) dx < +\infty, \] where $W^{1,n}_{0,{\rm rad}}(B)$ is the usual Sobolev spaces of radially symmetric functions on $B$ in $\mathbb R^n$ with $n\geq 2$. Without restricting to the class of functions $W^{1,n}_{0,{\rm rad}}(B)$, we should emphasize that the above inequalities fail in $W^{1,n}_{0,{\rm rad}}(B)$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.

math.AP↗

arXiv2019

A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in $\mathbb R^n$, $n\geq 3$, into the variable exponent Lebesgue space $L_{2^\star + |x|^α} (B)$, $2^\star = 2n/(n-2)$, $α>0$, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequality \[ \sup\Big\{\int_B |u(x)|^{2^\star+|x|^α} dx : u\in H^1_{0,{\rm rad}}(B), \|\nabla u\|_{L^2(B)} =1\Big\} < +\infty \] holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on $B$, namely, the embedding \[ H^m_{0,{\rm rad}}(B) \hookrightarrow L_{2_m^\star + |x|^α} (B) \] with $2\leq m < n/2$, $2_m^* = 2n/(n-2m)$, and $α>0$ holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.

math.AP↗

arXiv2019

Bubbling of the prescribed $Q$-curvature equation on $4$-manifolds in the null case

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\int_M f dμ_{{\it g}_0} <0$, then there exists at least one solution $u$ to the prescribed $Q$-curvature equation \[ \mathbf{P}_{g_0} u = f e^{4u}, \] where $\mathbf{P}_{g_0}$ is the Paneitz operator which is positive with kernel consisting of constant functions. In this paper, we fix a non-constant smooth function $f_0$ with \[ \max_{x\in M}f_0(x)=0, \quad \int_M f_0 dμ_{{\it g}_0} <0 \] and consider a family of prescribed $Q$-curvature equations \[ \mathbf{P}_{g_0} u=(f_0+λ)e^{4u}, \] where $λ>0$ is a suitably small constant. A solution to the equation above can be obtained from a minimizer $u_λ$ of certain energy functional associated to the equation. Firstly, we prove that the minimizer $u_λ$ exhibits bubbling phenomenon in a certain limit regime as $λ\searrow 0$. Then, we show that the analogous phenomenon occurs in the context of $Q$-curvature flow.

math.AP↗

arXiv2018

Higher order Sobolev trace inequalities on balls revisited

Inspired by a recent sharp Sobolev trace inequality of order four on the balls $\mathbb B^{n+1}$ found by Ache and Chang [AC15], we propose a slightly different approach to reprove Ache-Chang's trace inequality. To illustrate this approach, we reprove the classical Sobolev trace inequality of order two on $\mathbb B^{n+1}$ and provide sharp Sobolev trace inequalities of orders six and eight on $\mathbb B^{n+1}$. As the limiting case of the Sobolev trace inequality, a Lebedev-Milin type inequality of order up to eight is also considered.

math.AP↗

arXiv2017

Gradient estimates for some $f$-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces

Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $$ u_t=Δ_f u+au\log u+bu +Au^p+Bu^{-q} $$ and \[ u_t=Δ_f u+Ae^{pu}+Be^{-pu}+D \] are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.

math.DG↗

arXiv2017

A pointwise inequality for a biharmonic equation with negative exponent and related problems

Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order Hénon equation $Δ^2 u = |x|^a u^p$ in ${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly, Wei, and Xu [ Anal. PDE., 8(2015) 1541--1563], first for some positive constants $α$ and $β$ we establish the following pointwise inequality \[ Δu \geqslant αu^{-\frac{q-1}2} + βu^{-1} |\nabla u|^2 \] in ${\mathbb R}^n$ with $n \geqslant 3$ for positive $C^4$-solutions of the fourth-order equation \[ Δ^2u=-u^{-q} \quad \text{ in } \mathbb R^n \] where $q > 1$. Next, we prove a comparison property for Lane--Emden system with exponents of mixed sign. Finally, we give an analogue result for parabolic models by establishing a comparison property for parabolic system of Lane--Emden type. To obtain all these results, a new argument of maximum principle is introduced, which allows us to deal with solutions with high growth at infinity. We expect to see more applications of this new method to other problems in different contexts.

math.AP↗

arXiv2016

A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor

In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor $σ$ is small and the Yamabe invariant of the manifold is positive. The method we describe is however much simpler than the original method and allows easy extensions to several other problems. Some non-existence results are also considered.

gr-qc↗